AB2.15: Theorems of Real Analysis
|
|
- George Parsons
- 7 years ago
- Views:
Transcription
1 AB2.15: Theorems of Real Analysis Recall that a set of real numbers S R is called bounded if there is a real number M > 0 such that x < M for every x S. Definition 1. A function f : A B is called bounded provided there is a real number M > 0 such that f(x) < M for every x A. EXAMPLE 1 Show that the function f(x) = x 2 is unbounded on R but is bounded on each bounded interval I. Solution. If f(x) = x 2 is bounded on R then there exists a real number M > 0 such that x 2 < M for every x R; however, f(m + 1) = (M + 1) 2 > M + 1 > M, and this contradiction shows that f(x) = x 2 is bounded on R because there can be no bound for f(x) on R. If we restrict f(x) to a bounded interval I = [a, b] then f(x) M x [a, b], M = max{a 2, b 2 }, and, by Definition 1, f(x) = x 2 is bounded on I. A function f : A B is called bounded above if there is a real number M 1 such that f(x) M 1 for every x A. f : A B is called bounded below if there is a real number M 2 such that f(x) M 2 for every x A. Then a function f : A B is bounded if and only if it is bounded above and bounded below. A function f : A B is called bounded at x 0 A if there is an open interval J containing x 0 such that f is bounded on J A. If f is bounded on A then f is bounded at each x 0 A For example, f(x) = x 2 is bounded on every bounded interval I = [a, b] and hence is bounded at each real number x 0 THEOREM 1. If f(x) is bounded at each point x 0 A and A is a compact set then f(x) is bounded on A. In particular, if f(x) is bounded at each point x 0 [a, b] of a closed bounded interval then f(x) is bounded on [a, b]. Limit of a function Definition 2. The limit of f as x approaches a is a real number L, lim f(x) = L, provided that given any ɛ > 0 there is a δ > 0 such that if 0 < x a < δ then f(x) L < ɛ.
2 EXAMPLE 2 Show that lim(2x 1) = 5. x 3 Solution. Let ɛ > 0 be given. Find δ > 0 such that if 0 < x 3 < δ then (2x 1) 5 < ɛ, that is, x 3 < ɛ 2 Taking δ = ɛ 2, we observe that if 0 < x 3 < δ = ɛ 2 then 2 x 3 = 2x 6 = (2x 1) 5 < 2δ = ɛ. then THEOREM 2. If lim f(x) = L 1, lim g(x) = L 2 lim [f(x) ± g(x)] = L 1 ± L 2. lim f(x)g(x) = L 1L 2. lim 1/g(x) = 1/L 2, L 2 0. Prove the first equality. The limit of f as x approaches a is a number L 1, provided that given any ɛ > 0 there exists a δ 1 > 0 such that if 0 < x a < δ 1 then f(x) L 1 < ɛ/2. Similarly, the limit of g as x approaches a is a number L 2, provided that given any ɛ > 0 there exists a δ 2 > 0 such that if 0 < x a < δ 2 then g(x) L 2 < ɛ/2. Take δ = min{δ 1, δ 2 }; then both inequalities f(x) L 1 < ɛ/2 and g(x) L 2 < ɛ/2 hold if 0 < x a < δ. So [f(x) ± g(x)] (L 1 ± L 2 ) = [f(x) L 1 ] ± [g(x) L 2 ] f(x) L 1 + g(x) L 2 < ɛ/2 + ɛ/2 = ɛ. Continuity Definition 3. A function f(x) is called continuous at x 0 if lim f(x) = f(x 0 ). x x 0 Equivalently, given any ɛ > 0 there is a δ > 0 such that if x a < δ then f(x) f(x 0 ) < ɛ.
3 Definition 3 can be summarized in three stages: (i) x 0 is in the domain of f; that is, f(x 0 ) exists; (ii) x x0 lim f(x) exists; (iii) x x0 lim f(x) and f(x 0 ) are equal. If any of these three conditions fails, f is not continuous at x 0. We say that a function f(x) is discontinuous at x 0 or has a discontinuity (break) at x 0. If condition (ii) holds but (i) (and hence (iii)) fails, or if (ii) and (i) hold but (iii) fails then f is said to have a removable discontinuity at x 0. EXAMPLE 3 Prove that f(x) = x2 4 x 2 has a removable discontinuity at x 0 = 2. Solution. The function f(x) = (x 2 4)/(x 2) = x + 2, x 2 is not defined at x 0 = 2 but it is defined at every other real number and hence in a deleted neighborhood 0 < x 2 < δ of x = 2. Hence x = 2 is not in the domain of f and f(2) does not exist. On the other hand lim f(x) = lim x 2 x 2 (x2 4)/(x 2) = lim(x + 2) = 4, x 2 which means that f has a removable discontinuity at x 0 = 2: if we define f(2) = 4 then f(x) has been made continuous at x = 2. EXAMPLE 4 Prove that has a jump discontinuity at x = 0 and f(x) = sgn x, f(0) = 0 f(x) = sgn x, f(0) = 0 has a removable discontinuity at x = 0. Solution. The function f(x) = sgn x, f(0) = 0 is defined at x 0 = 0 and at every other real number and hence in a deleted neighborhood 0 < x < δ of x = 0. However, lim f(x) = 1, lim x +0 f(x) = 1 x 0 which means that lim x ±0 f(x) and f(0) are not equal and lim x 0 f(x) does not exist. Therefore f(x) = sgn x is not continuous and has a jump discontinuity at x = 0 with a jump 2 because
4 lim f(x) lim f(x) = 2. x +0 x 0 On the other hand, f(x) = sgn x has a removable discontinuity at x 0 = 2: if we define f(0) = 1 then f(x) has been made continuous at x = 1 because lim x +0 f(x) = lim x 0 f(x) = lim x 0 f(x) = f(0) = 1. Definition 4. A function f(x) has a simple discontinuity (discontinuity of the first kind) at x = x 0 if the discontinuity is either removable or a jump discontinuity. Every other discontinuity is called a discontinuity of the second kind. So if both limits lim f(x) and lim f(x) exist but x +x 0 x x 0 lim f(x) lim f(x), x +x 0 x x 0 then f(x) has a jump discontinuity. If at least one of the two one-sided limits at x = x 0 fails to exist, f(x) has a discontinuity of the second kind. THEOREM 3. If f(x) and g(x) are each continuous at x = x 0 then f ± g, fg are continuous at x = x 0 and f/g is continuous at x 0 if g(x 0 ) 0. Prove the continuity for f ± g. For f(x) and g(x) that are continuous at x = x 0, we have Then, by Theorem 2, lim f(x) = L 1 = f(x 0 ), lim g(x) = L 2 = g(x 0 ). which proves the continuity of f ± g. Thus, every polynomial lim [f(x) ± g(x)] = L 1 ± L 2 = f(x 0 ) ± g(x 0 ), f(x) = a 0 x n + a 1 x n a n 1 x + a n is continuous at each real number x = x 0, and every rational function f(x) = a 0x n + a 1 x n a n 1 x + a n b 0 x m + b 1 x m b m 1 x + b m (a ratio of two polynomials) is continuous at each real number x = x 0 except the finitely many zeros of the denominator polynomial q(x) = b 0 x m + b 1 x m b m 1 x + b m. THEOREM 4. If f(x) is continuous at x 0 and g(x) is continuous at f(x 0 ) then g(f(x)) is continuous at x = x 0. PROOF. Let ɛ > 0 be given. Since g(x) is continuous at f(x 0 ) there is a η > 0 such that if y f(x 0 ) < η then g(y) g(f(x 0 )) < ɛ. But since f(x) is continuous at x 0 there is a δ > 0 such that if x x 0 < δ then f(x) f(x 0 ) < η. Consequently, for each x satisfying x x 0 < δ we have g(f(x)) g(f(x 0 )) < ɛ. Therefore g(f(x)) is continuous at x = x 0.
5 EXAMPLE 5 Prove that f(x) = sin 1 x has a discontinuity of the second kind at x = 0. Solution. Consider the function f(x) = sin(1/x) and define f(0) = 0. The rational function 1/x is continuous at each nonzero real number and the function sin x is continuous at each real number. Hence, by Theorem 4, f(x) is continuous at each x 0. By considering the sequence we see that x n = 2 2πn 0 lim f(x) x +0 does not exist and so f(x) has a discontinuity of the second kind at x = 0. Definition 5. A function f(x) is called right-continuous at x 0 if or left-continuous at x 0 if lim x +x 0 f(x) = f(x 0 ). lim x x 0 f(x) = f(x 0 ). Thus f(x) is continuous at x 0 if and only if it is both left- and right-continuous at x 0. Properties of continuous functions LEMMA 1. If f(x) is continuous at x 0 then there exists a δ > 0 such that f is bounded on (x 0 δ, x 0 + δ); that is f(x) is bounded at x 0. PROOF. Since f(x) is continuous at x 0 there is a δ > 0 such that if x x 0 < δ then f(x) f(x 0 ) < 1, so that f(x) < 1 + f(x 0 ) = M for each x (x 0 δ, x 0 + δ). Therefore f(x) is bounded at x 0. THEOREM 5. If f(x) is continuous on the closed bounded interval [a, b] then f(x) is bounded on [a, b]. PROOF. Suppose f(x) : [a, b] R is continuous on the closed bounded interval [a, b]. By Lemma 1 f(x) is bounded at each x 0 [a, b]; thus f(x) is bounded on [a, b] by Theorem 1. COROL- LARY. If f(x) is continuous on R then f(x) is bounded on every bounded itnterval [a, b]. Define M = sup x A f(x), m = inf x A f(x).
6 EXAMPLE 6 Let f(x) = x 2 on (0, 2). f(x) is continuous and bounded on a bounded interval (0, 2). M = 4 and m = 0 but there are no points x 1, x 2 (0, 2) with f(x 1 ) = 4 and f(x 2 ) = 0. EXAMPLE 7 Let f(x) = x x { x x = if x > 0, 1+x 2 2 x2 if x < 0, 1+x 2 on R. f(x) is continuous and bounded on R. M = 1 and m = 1 (f is an odd function) but there are no points x 1, x 2 R with f(x 1 ) = 1 and f(x 2 ) = 1. THEOREM 6 (extreme-value theorem). If f(x) is continuous on the closed bounded interval [a, b] then there exist points x 1, x 2 [a, b] such that f(x 2 ) f(x) f(x 1 ) for all x [a, b]. That is, f(x 1 ) = M, f(x 2 ) = m. PROOF. If f(x) is continuous on the closed bounded interval [a, b] then, by Theorem 5, f(x) is bounded on [a, b], and so M = sup x A f(x), m = inf x A f(x) exist as real numbers. Suppose that value M is not assumed; then f(x) < M x [a, b]. Define g(x) = 1 M f(x), x [a, b]. Clearly, g(x) > 0 x [a, b] and g(x) is continuous on [a, b]. Then, by Theorem 5, g(x) is bounded on [a, b], and so there is a real number k > 0 such that g(x) k x [a, b]. Now for every x [a, b] k g(x) = 1 M f(x), and so M f(x) 1 k > 0, f(x) M 1, x [a, b], k and this contradicts the definition of M as the least upper bound of f(x) on [a, b]. Therefore, value M must be assumed; i.e., there exist a point x 1 [a, b] with f(x 1 ) = M.
7 A similar statement for m and x 2 [a, b] can be proved by applying the same argument to f. THEOREM 7 (intermediate-value theorem). If f(x) is continuous on the closed bounded interval [a, b] and k is between f(a) and f(b) then there exist a c [a, b] such that f(c) = k. PROOF. Let f(x) be continuous on the closed bounded interval [a, b] and k be between f(a) and f(b); that is, either f(a) < k < f(b) or f(b) < k < f(a). We will prove the former. Define the function g(x) = f(x) k, x [a, b]. Then Let g(a) = f(a) k < 0, g(b) = f(b) k > 0, x [a, b]. c = sup {x [a, b] : g(x) < 0}. Since g(x) is continuous on the closed bounded interval [a, b] then g(x) < 0 on some interval [a, a + δ 1 ], δ 1 > 0 and g(x) > 0 on some interval [b δ 2, b], δ 2 > 0. Thus a < c < b. Now for every x satisfying c < x b g(x) 0, for otherwise c would not be an upper bound of the set {x [a, b] : g(x) < 0}. Hence By continuity of g, lim g(x) 0. x +c g(c) = lim g(x) : g(c) 0. x +c Choose a sequence x n in [a, b] such that x n < c and g(x n ) < 0 for every integer n. Note that if there no such a sequence then the set {x [a, b] : g(x) < 0} would have an upper bound less than c, in contradiction with the definition of c. Now since g(x) is continuous at c, g(x n ) g(c). Hence g(c) 0. It follows that g(c) = 0 and consequently f(c) = k. EXAMPLE 8 Prove that the function f(x) = x + 1, x (0, 1], f(0) = 0 does not satisfy the conclusion of the intermediate-value theorem.
8 Solution. f is bounded but not continuous on [0, 1] since f fails to be right-continuous at x = 0. We have M = sup x [0,1] f(x) = 2, m = inf x [0,1] f(x) = 0. In fact f(0) = 0 and f(1) = 2 but there is no c (0, 1) with f(c) = 1, for example. It is easy to see that none of the intermediate values x (0, 1] are assumed by f. THEOREM 8. If f(x) is continuous on the closed bounded interval [a, b] and f(x) [a, b] then f has a fixed point: there exists and x 0 [a, b] such that f(x 0 ) = x 0. PROOF. Let f(x) be continuous on the closed bounded interval [a, b] and f(x) [a, b] for every and x [a, b]. If f(a) = a or f(b) = b then the theorem is proved; hence assume that f(a) < a and f(b) < b. Define g(x) = f(x) x for every and x [a, b]. Clearly g(a) > 0a, g(b) < 0, and g(x) is continuous on [a, b]. Now 0 is an intermediate value for g on [a, b]. Now, by the intermediate-value theorem, there exist a c = x 0 [a, b] such that g(x 0 ) = 0. Then f(x 0 ) = x 0. THEOREM 9. If f(x) is one-to-one and continuous on the closed bounded interval [a, b] then f is strictly monotone on [a, b]. PROOF. Let f(x) be one-to-one and continuous on the closed bounded interval [a, b]. Then f(a) f(b). Consider the case f(a) < f(b). If f is not strictly increasing on [a, b] then there exists x 1, x 2 [a, b] such that x 1 < x 2, f(x 1 ) f(x 2 ). Equality here violates the assumption that f(x) is one-to-one and continuous, and so we must have f(x 1 ) > f(x 2 ). There are two possibilities: (a) f(x 1 ) > f(b). Choose k (f(b), f(x 1 )). Then k (f(a), f(x 1 )), and by the intermediatevalue theorem there is a c 1 (a, x 1 ) with f(c 1 ) = k and there is a c 2 (x 1, b) with f(c 2 ) = k. But c 1 c 2 and this contradicts the assumption that f(x) is one-to-one. (b) f(x 1 ) < f(b). Then f(x 2 ) < f(b). Choose k (f(x 2 ), f(x 1 )). Then k (f(x 2 ), f(b)), and by the intermediate-value theorem there is a c 1 (x 1, x 2 ) with f(c 1 ) = k and there is
9 a c 2 (x 2, b) with f(c 2 ) = k. But c 1 c 2 and this contradicts the assumption that f(x) is one-to-one. It follows that f is strictly increasing on [a, b]. Finally for the case f(a) > f(b) the above argument applied to f shows that f is strictly increasing on [a, b] and hence f is strictly decreasing on [a, b]. [a, b]. THEOREM 10. If f(x) is one-to-one and continuous on [a, b] then f 1 is continuous on [m, M] where M = sup x [a,b] f(x), m = inf x [a,b] f(x). PROOF. We know that the range of f 1 is the closed bounded interval [m, M] and so f 1 is a well-defined one-to-one function defined on [m, M] with the range [a, b]. Let y 0 [m, M]. It suffices to prove that f 1 is continuous at y 0. Choose a sequence y n in [m, M] such that y n y 0. It is sufficient to show that Let f 1 (y n ) f 1 (y 0 ), n. x 0 = f 1 (y 0 ), x n = f 1 (y n ) n = 1, 2,..., and suppose that x n does not converge to x 0. Then for some ɛ > 0 x n x 0 ɛ for infinitely many integers n. Extract a subsequence x n of x n such that x n x 0 ɛ n. Now x n is a bounded sequence and by the Bolzano Weierstrass theorem x n has a convergent subsequence, say x n. Then By the continuity of f at c, x n x 0 ɛ n, x n c [a, b], c x 0. f( x n) f(c). But {f( x n)} is a subsequence of {f(x n ) = y n } and y n y 0 = f(x 0 ). Therefore, f(c) = f(x 0 ). Since this contradicts the assumption that f is a one-to-one function, we must have x n x 0 ; that is, f 1 (y n ) f 1 (y 0 ), n, which means that f 1 is continuous on [m, M]. Derivative. Differentiable functions Definition 6. The derivative of a function f(x) at x 0 is f (x 0 ) = lim f(x) f(x 0 + h) f(x 0 ). h 0 h
10 Whem the limit exists we say that f(x) is differentiable at x 0. or THEOREM 11. If f(x) is differentiable at x 0 then f(x) is continuous at x 0. PROOF. Recall that a function f(x) is continuous at x 0 if Assume that f(x) is differentiable at x 0. Then lim [f(x) f(x 0 )] = lim x x 0 f(x) f(x 0 ) x x0 x x 0 lim f(x) = f(x 0 ), x x 0 lim [f(x) f(x 0 )] = 0. x x 0 (x x 0 ) = lim x x0 f(x) f(x 0 ) x x 0 x x0 lim (x x 0 ) = f (x 0 ) 0 = 0. THEOREM 12 If f(x) has a local extremum (maximum or minimum) then either f (x 0 ) = 0 or f (x 0 ) does not exist. PROOF Let f(x) have a local maximum. We have If f (x 0 ) exists then necessarily By the above f(x) f(x 0 ) x x 0 0, x 0 < x < x 0 + δ. f(x) f(x 0 ) x x 0 0, x 0 δ < x < x 0. f(x) f(x 0 ) lim (x x 0 ) = f f(x) f(x 0 ) (x 0 ) = lim x +x 0 x x 0 x x 0 x x 0 f(x) f(x 0 ) f(x) f(x 0 ) lim (x x 0 ) 0, lim 0. x +x 0 x x 0 x x 0 x x 0 It follows that f (x 0 ) = 0. If f (x 0 ) does not exist, we have several obvious examples when f(x) has a local extremum. Note that the converes is not true: If f (x 0 ) = 0 or f (x 0 ) does not exist then f(x) may not have a local extremum (maximum or minimum). To see this consider the function f(x) = x 3 at x = 0. THEOREM 13 (Rolle s theorem). If f(x) is continuous on [a, b] and differentiable on (a, b) and f(a) = f(b) then there exists an x 0 (a, b) such that f (x 0 ) = 0. PROOF If f(x) = f(a) for every x (a, b) then f is constant on [a, b] and so f (x) = 0 for every x (a, b). Hence we can assume that there is some x (a, b) for which f(x) f(a). By
11 the extreme-value theorem, f assumes its absolute-maximum and absolute-mainimum values on [a, b]; that there exists x 1, x 2 [a, b] such that f(x 1 ) f(x) f(x 2 ) x [a, b]. By our assumption that f is not constant on [a, b] together with the fact that f(a) = f(b) it follows that there is x 0 (a, b) such that f has an absolute extremum at x 0, and f(x 0 ) is a local extremum of f. Also f (x 0 ) exists by the condition that f(x) is differentiable on (a, b). Thus, by Theorem 12, f (x 0 ) = 0. THEOREM 14 (Cauchy mean-value theorem). If f(x) and g(x) are each continuous on [a, b] and differentiable on (a, b) then there exists an x 0 (a, b) such that f (x 0 )[g(b) g(a)] = g (x 0 )[f(b) f(a)]. PROOF Let F (x) = f(x)[g(b) g(a)] g(x)[f(b) f(a)]. F is continuous on [a, b] and differentiable on (a, b). Also F (b) = f(b)[g(b) g(a)] g(b)[f(b) f(a)] = f(a)g(b) g(a)f(b) = f(a)[g(b) g(a)] g(a)[f(b) f(a)] = F (a) By Rolle s theorem, there exists an x 0 (a, b) such that F (x 0 ) = 0. But F (x) = f (x)[g(b) g(a)] g (x)[f(b) f(a)]. Hence f (x 0 )[g(b) g(a)] = g (x 0 )[f(b) f(a)].
12 PROBLEM 15.1 Show that the function f(x) = 1/x is unbounded for 0 < x < but is bounded on (a, ) for every a > 0. Solution. Take M > 0 and x > 0 such that x < 1/M; then 1/x > M so that M is not a bound for f(x) and, by Definition 1, f(x) = 1/x is unbounded for 0 < x <. Note that f(x) = 1/x is bounded at each x 0 (0, 1) (and at each x 0 > 0) but is not bounded on (0, 1). PROBLEM 15.2 Show that the function is bounded for all real x. Solution. Since we have f(x) = sin x, x 0, f(0) = 1 x sin x x x R, f(x) = sin x x and we can choose, e.g., M = 2 as a bound for f(x) so that, by Definition 1, f(x) = sin x/x is bounded on R. Note that 1/x is unbounded on R and sin x is bounded on R; however, their product, f(x) = sin x/x, is bounded on R (and at each point x 0 ). PROBLEM 15.3 Show that the function f(x) = x sin x 1, is unbounded for all real x (on R). Solution. Take an arbitrary M > 0 and choose integers k 1 and k 2 such that 2πk 1 + π 2 > M, 2πk 2 + π 2 < M. Now, Hence, sin(2πk 1 + π 2 ) = 1, sin(2πk 2 π 2 ) = 1, f(2πk 1 + π 2 ) = 2πk 1 + π 2 > M, f(2πk 2 π 2 ) = 2πk 2 + π 2 < M. Thus f(x) is neither bounded above nor bounded below and thus unbounded on R. PROBLEM 15.4 Prove that x 2 4 lim x 2 x 2 = 4.
13 Solution. The function f(x) = x2 4 x 2 = (x 2)(x + 2) x 2 = x + 2, x 2 is not defined at x = 2 but it is defined at every other real number and hence in a deleted neighborhood 0 < x 2 < δ of x = 2. Let ɛ > 0 be given. Find δ > 0 such that if 0 < x 2 < δ then f(x) 4 < ɛ, that is, Taking, for example (or δ = ɛ/2) we observe that if then PROBLEM 15.5 (x + 2) 4 = x 2 < ɛ (x 2). δ = ɛ 0 < x 2 < δ ɛ x 2 = (x + 2) 4 = f(x) 4 < δ ɛ. Prove that lim x 1 x 2x + 1 = 1. Solution. The function f(x) = x 2x + 1 is defined at every real number x 1/2 and hence in a deleted neighborhood 0 < x+1/2 < δ of x = 1/2 and in a deleted neighborhood 0 < x+1 < η of x = 1 provided that 0 < η 1/2. Let ɛ > 0 be given. Find δ > 0 such that if 0 < x + 1 < δ then f(x) 1 < ɛ. That is, we must prove that for such δ Assume that x 2x x 1 = 2x + 1 = x x + 1 < ɛ x + 1 < 1 ( 4 : x 5 ) 4, 3. 4 (x 1/2). Then 2x + 1 > 2x 1 > = 1 2,
14 so that Hence which means that for Taking 1 2x + 1 < 2. 1 f(x) 1 = x + 1 < 2 x + 1 2x + 1 f(x) 1 < ɛ 0 < x + 1 < δ = ɛ 2 δ = min{ ɛ 2, 1 4 } we observe that both inequalities x + 1 < 1 4 Note that the initial restriction and f(x) 1 < ɛ will be satisfied. x + 1 < 1 4 could be replaced by for any fixed positive x + 1 < η η < 1 2. PROBLEM 15.6 Prove that Solution. The function lim x = a, a > 0. f(x) = x is defined at every real number x 0 and hence in a deleted neighborhood 0 < x a < δ of x = a > 0. Now x + a > a so that Take ɛ > 0 and we obtain x x a a = < x a 1 x + a a δ = min{a, ɛ a} f(x) a = x a < x a 1 a < δ a ɛ a a = ɛ. PROBLEM 15.7
15 Prove that f(x) = x sin 1 x has a removable discontinuity at x = 0. Solution. Consider the function f(x) = x sin(1/x) and define f(0) = 0. The rational function 1/x is continuous at each nonzero real number and the functions sin x and x are continuous at each real number. Hence, by Theorem 3, f(x) is continuous at each x 0. The limit below exists, and lim f(x) = 0 x +0 because sin x 1 and lim x = 0. Therefore we have x 0 (i) f(0) defined as f(0) = 0 exists; (ii) lim f(x) exists; x 0 (iii) lim f(x) and f(0) are equal (to zero). x 0 So f(x) has a removable discontinuity at x = 0 and is continuous at x = 0, (and thus at each real number). PROBLEM 15.8 Find and classify all discontinuity points of the function f(x) = x 2 if x < 1, 2x + 3 if 1 x 0, x 1 if 0 < x < 2, x 3 7 if 2 x < 3, (x 3)/(x 4) if 3 x < 4, 0 if x 4, PROBLEM 15.9 Find for M = sup x [a,b] f(x), m = inf x [a,b] f(x) x 2 + 2x + 3 if x [0, 4], 2 x 1 if x [ 2, 2], f(x) = e 1/x if x [0, ), 1 x 2 if x ( 2, 1], PROBLEM Prove that the polynomial p(x) = a 0 x n + a 1 x n a n 1 x + a n, a 0 0, n = 2l 1, l = 1, 2,...,
16 has at least one real root x 0 R: p(x 0 ) = 0. Solution. Consider, for example We have p(x) = x 3 + 2x 2 + x + 1. p( 3) = = 11 < 0, p( 1) = = 1 > 0. Thus, by the intermediate-value theorem, there is an x 0 ( 3, 1) at which p(x 0 ) = 0 because 11 < 0 < 1. For a polynomial p(x) = a 0 x n + a 1 x n a n 1 x + a n, a 0 > 0, n = 2l 1 (l = 1, 2,...), one can choose a sufficiently large by modulus negative x 1 such that a 0 x n > a 1 x n a n 1 x + a n, x < x 1 and a sufficiently large positive x 2 such that a 0 x n > a 1 x n a n 1 x + a n, x > x 2. Thus, by the intermediate-value theorem, there is an x 0 (x 1, x 2 ) at which p(x 0 ) = 0.
1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationMA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationContinuity. DEFINITION 1: A function f is continuous at a number a if. lim
Continuity DEFINITION : A function f is continuous at a number a if f(x) = f(a) REMARK: It follows from the definition that f is continuous at a if and only if. f(a) is defined. 2. f(x) and +f(x) exist.
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More informationReal Roots of Univariate Polynomials with Real Coefficients
Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials
More informationRolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More information5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
More informationLecture Notes on Analysis II MA131. Xue-Mei Li
Lecture Notes on Analysis II MA131 Xue-Mei Li March 8, 2013 The lecture notes are based on this and previous years lectures by myself and by the previous lecturers. I would like to thank David Mond, who
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More information5.3 Improper Integrals Involving Rational and Exponential Functions
Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationTwo Fundamental Theorems about the Definite Integral
Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than
More informationCHAPTER 7. Di erentiation
CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.
More informationChapter 7. Continuity
Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this
More informationMetric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationLecture Notes for Analysis II MA131. Xue-Mei Li with revisions by David Mond
Lecture Notes for Analysis II MA131 Xue-Mei Li with revisions by David Mond January 8, 2013 Contents 0.1 Introduction............................ 3 1 Continuity of Functions of One Real Variable 6 1.1
More information3.3 Real Zeros of Polynomials
3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationMATH PROBLEMS, WITH SOLUTIONS
MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These
More informationLimits and Continuity
Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function
More information1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationThe Division Algorithm for Polynomials Handout Monday March 5, 2012
The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationGod created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)
Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work
More informationcorrect-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationTo define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions
Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More information1. Let X and Y be normed spaces and let T B(X, Y ).
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationNonlinear Algebraic Equations. Lectures INF2320 p. 1/88
Nonlinear Algebraic Equations Lectures INF2320 p. 1/88 Lectures INF2320 p. 2/88 Nonlinear algebraic equations When solving the system u (t) = g(u), u(0) = u 0, (1) with an implicit Euler scheme we have
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationWinter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com
Polynomials Alexander Remorov alexanderrem@gmail.com Warm-up Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).
More informationPutnam Notes Polynomials and palindromes
Putnam Notes Polynomials and palindromes Polynomials show up one way or another in just about every area of math. You will hardly ever see any math competition without at least one problem explicitly concerning
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationsin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2
. Problem Show that using an ɛ δ proof. sin() lim = 0 Solution: One can see that the following inequalities are true for values close to zero, both positive and negative. This in turn implies that On the
More information3.6 The Real Zeros of a Polynomial Function
SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,
More information1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]
1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not
More informationON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction
ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More informationDomain of a Composition
Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationMath 104: Introduction to Analysis
Math 104: Introduction to Analysis Evan Chen UC Berkeley Notes for the course MATH 104, instructed by Charles Pugh. 1 1 August 29, 2013 Hard: #22 in Chapter 1. Consider a pile of sand principle. You wish
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is
More informationCalculus. Contents. Paul Sutcliffe. Office: CM212a.
Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical
More informationThe Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line
The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationHow To Prove The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More informationCHAPTER 2: METHODS OF PROOF
CHAPTER 2: METHODS OF PROOF Section 2.1: BASIC PROOFS WITH QUANTIFIERS Existence Proofs Our first goal is to prove a statement of the form ( x) P (x). There are two types of existence proofs: Constructive
More informationLectures 5-6: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationIntermediate Value Theorem, Rolle s Theorem and Mean Value Theorem
Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem February 21, 214 In many problems, you are asked to show that something exists, but are not required to give a specific example or formula
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More information4.3 Lagrange Approximation
206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationGENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY
GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY CARL G. JOCKUSCH, JR. AND PAUL E. SCHUPP Abstract. Generic decidability has been extensively studied in group theory, and we now study it in
More informationf(x) = g(x), if x A h(x), if x B.
1. Piecewise Functions By Bryan Carrillo, University of California, Riverside We can create more complicated functions by considering Piece-wise functions. Definition: Piecewise-function. A piecewise-function
More information