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1 Continuity Continuity is function chrcteristic. Some functions re continuous over the entire number line,. Others re continuous only for some proper subset of the number line. Continuity implies n bsence of interruptions. If function is continuous on n intervl its grph is unbroken on the intervl. If function is not continuous, its grph is broken nd sid to be discontinuous. Formlly, continuity requires three conditions. If function fils condition one or condition two t some point, then the function lso fils condition three t tht point. The three grphs below ech demonstrte filure to stisfy t lest two of the three conditions for continuity t point. Qx ( ) fils condition one, for the function is not defined t. (It lso fils condition two since the limit s x pproches does not exist.) Qx ( ) exhibits type of discontinuity clled infinite discontinuity. J( x ) stisfies condition one but fils condition two t point since the limit s x pproches does not exist. J( x ) exhibits step or jump discontinuity so clled becuse the function "steps" or "jumps" from one vlue to nother. R( x ) stisfies conditions one nd two, but not condition three t point becuse ( ) ( ) A function f is continuous t number if 1. f ( ) is defined, 2. lim f x ( x) exists, 3. lim ( ) ( ) f x = f. lim R x R. R( x ) exhibits removble discontinuity so clled becuse the function could be mde continuous by redefining it t single point. Q( x) J ( x) R ( x) Due to the relince on limits, the definition of continuity simultneously stresses single point nd n intervl of points. Ambivlence bounds: continuity is point property tht occurs within n intervl of defined vlues of f. Consider the grph of f below. The grph exhibits brek or interruption when x =. We sy, "Function f is discontinuous t x =." The limit of f s x pproches equls L, but f ( ) = M. Since lim f ( x x ) f ( ), the third condition for continuity is not met.

2 f ( x) L M i To describe the intervls of continuity, we write (, ) (, ). Notice the use of prenthesis with the number. There must be n emphsis on the open intervl becuse f is not continuous t. A function is continuous on the open intervl (,b) if it is continuous t every point in the closed intervl [,b] except possibly t the endpoints nd b. Consider the grph of ρ below. The grph exhibits brek or interruption when x = 0. We sy, "Function ρ is discontinuous t x = 0." The limit of ρ s x pproches zero does not exist, so the function fils condition two when x = 0. ρ Despite the discontinuity t x = 0, we cn sy, "Function ρ is continuous on the open,0 0,." Intuitively, there seems to be difference between the intervl intervls ( ) ( ) (,0) versus the intervl ( ) 0,. To drw distinction between these two intervls, we will pply the following definitions.

3 A function f is continuous from the right t number if lim + x ( ) ( ) f x = f, nd f is continuous from the left t number if lim x ( ) ( ) f x = f. Now, we cn del with functions like ρ on the intervl [ 0, ). We will sy, "Function ρ is continuous on the intervl [ 0, )." To mke this sttement, we need the definition below. A function f is continuous on n intervl [,b) if it is continuous t every number within the intervl (,b) nd is either continuous or continuous from the right t. Similrly, function f is continuous on n intervl (,b] if it is continuous t every number within the intervl (,b) nd is either continuous or continuous from the left t b. To estblish the continuity of function over n intervl using the definitions from bove, it is necessry to estblish tht the function is continuous t n infinite number of x- vlues, dunting tsk to sy the lest. Accordingly, we will estblish the continuity of some functions by ccepting few theorems. Some functions re continuous over the entire number line. -Continuous Theorem: Exponentil functions, sine-wve nd cosine-wve,. functions, nd polynomil functions re continuous over the intervl ( ) Some functions tht not continuous over the number line re continuous over their domin. D-Continuous Theorem: Root functions, rtionl functions, logrithmic functions, nd the remining trigonometric functions re ll continuous over their domins. Functions derived from opertions with continuous functions re themselves continuous. Continuity with Function Opertions Theorem: If f nd g re continuous t, then the following functions re continuous t : 1. f ± g 2. fg f 3. g if g( ) 0 4. c f where c is constnt.

4 Using the bove theorems, we cn show tht ( sin ) numbers except zero. Let f ( x) = sin x nd g( x) y = x x is continuous for ll rel = x. Function f is the sine-wve function, which is continuous by the - Continuous Theorem. Function g is first-degree polynomil function, which is lso continuous by the - Continuous Theorem. By the Continuity with Function Opertions Theorem, we know sin x x is continuous for ll x except zero. Prctice Problems 1st ed. problem set: 2nd ed. problem set: 3rd ed. problem set: 2.4 #1 5 odd, #11 17 odd 2.4 #1 5 odd, #9, #13 19 odd 2.4 #1 5 odd, #9, #13 19 odd Possible Exm Problems #1 Wht re the three conditions for continuity t point. Answer: 1. f ( ) is defined, 2. lim ( ) f x exists, 3. lim ( ) ( ) f x = f. L x x #2 Consider ( ) = + 1. Is L(x) continuous on the following intervls? I. [ 3,3] II. [ 1, ) III. IV. [ 0,17.3 ] Answer: I. No. II. Yes. III. No. IV. Yes. #3 Given f ( x). x =, wht conditions cn be plced on n such tht f will be continuous over n 2 + x Answer: If n is positive even integer, then f is continuous over. #4 Which condition(s) of continuity re not met by β ( x) whose grph is given below. Answer: Condition three is not met. β is discontinuous becuse lim ( ) ( ) f x f.

5 Exmple Exercise 1 Consider the grph of f ( x ) below. Stte the conditions of continuity tht re not met t ech discontinuous point within the intervl The grph shows tht the curve is discontinuous t x = 2 where the function fils to f 2 lim f x. meet the definition of continuity, nmely, ( ) ( ) x 2 The grph shows tht the curve is discontinuous t 1 x = becuse lim f ( x) x 1 does not exist nd becuse f ( 1) is undefined. For the sme resons, the curve is discontinuous t x = 1.

6 Exmple Exercise 2 Sketch function f such tht f hs the following chrcteristics. i) f is discontinuous when x = 5, x = 3, nd x = 2 f 5 = 1 ii) ( ) iii) f ( x) lim = 2 x 5 iv) f ( 3 ) is undefined v) lim f ( x) = x 3 vi) lim f ( x) = 0 x Sketches my vry, but the following grph exhibits ll the given chrcteristics.

7 Appliction Exercise The function F ( r ) gives the grvittionl force exerted by Erth on unit mss t distnce r from the center of the plnet. GMr 3 R F( r) = GM 2 r if r < R if r R In the formul, M, R, nd G re non-zero constnts representing the mss of Erth, the rdius of Erth, nd the grvittionl constnt respectively. Is F D-continuous? In other words, is F continuous for non-zero vlues of r?