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1 Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion b Continuous and nowhere differentiable c Differentiable and nowhere monotone Hints to exercises a Finite Taylor expansion An infinitely differentiable function R R need not be analytic. It has a formal Taylor expansion, but maybe of zero radius of convergence, or maybe converging to a different function. An example: f(x) = e 1/x for x > 0, f(x) = 0 for x 0. 9a1 Theorem. 1 2 If an infinitely differentiable function f : R R is not a polynomial then there exists x R such that f (n) (x) is irrational for all n. Thus, x n f (n) (x) 0. The set of rational numbers may be replaced with any other countable set. We ll prove the theorem via iterated Baire category theorem. 9a2 Lemma. If f is a polynomial on [a, b] and n f(b + ε n ) = f(b) for some ε n 0+ then f is constant on [a, b]. 1 Exercise in book: B. Thomson, J. Bruckner, A. Bruckner, Real analysis, second edition, The theorem: Theorem: Let f(x) be C on (c, d) such that for every point x in the interval there exists an integer N x for which f (Nx) (x) = 0; then f(x) is a polynomial. is due to two Catalan mathematicians: F. Sunyer i Balaguer, E. Corominas, Sur des conditions pour qu une fonction infiniment dérivable soit un polynôme. Comptes Rendues Acad. Sci. Paris, 238 (1954), F. Sunyer i Balaguer, E. Corominas, Condiciones para que una función infinitamente derivable sea un polinomio. Rev. Mat. Hispano Americana, (4), 14 (1954). The proof can also be found in the book (p. 53): W. F. Donoghue, Distributions and Fourier Transforms, Academic Press, New York, I will never forget it because in an Exercise of the Opposition to became Full Professor I was posed the following problem: What are the real functions indefinitely differentiable on an interval such that a derivative vanish at each point? Juan Arias de Reyna; see Question on Mathoverflow.

2 Tel Aviv University, 2013 Measure and category 76 Proof. We have f (n) (b) = 0 for n = 1, 2,... since otherwise f(b + ε) = f(b) + cε k + o(ε k ) for some k 1 and c 0. The same holds for f(a ε n ), of course. Assume that f is a counterexample to Theorem 9a1. Consider a (maybe empty) set P f of all maximal nondegenerate intervals I R such that f is a polynomial on I. Note that intervals of P f are closed and pairwise disjoint. 9a3 Lemma. The open set G f = I P f Int I is dense (in R). Proof. Closed sets F n,r = {x : f (n) (x) = r} for r Q and n = 0, 1, 2,... cover R. By (5b7), n,r Int F n,r is dense. Clearly, f is a polynomial on each interval contained in this dense open set. It follows that P f, treated as a totally (in other words, linearly) ordered set, is dense (that is, if I 1, I 2 P f, I 1 < I 2 then I P f I 1 < I < I 2 ). It may contain minimal and/or maximal element (unbounded intervals), but the rest of P f, being an unbounded dense countable totally ordered set, is order isomorphic to Q (0, 1) (the proof is similar to the proof of Lemma 2d4; so-called back-and-forth method). Now we want to contract each interval of P f into a point. (We could consider a topological quotient space... ) We take an order isomorphism ϕ : P f Q between P f and one of Q (0, 1), Q [0, 1), Q (0, 1], Q [0, 1], and construct an increasing ψ : R [0, 1] such that ψ(x) = ϕ(i) whenever x I. Clearly, such ψ exists and is unique. It is continuous. The image ψ(r) is one of (0, 1), [0, 1), (0, 1], [0, 1]. In every case ψ(r) is completely metrizable. Note that ψ 1 (Q) = I Pf I, and ψ is one-to-one on R \ I Pf I. We define E n,r ψ(r) for r Q and n = 0, 1, 2,... as follows: E n,r = {x : ψ 1 (x) F n,r }. 9a4 Lemma. Each E n,r is closed in ψ(r).

3 Tel Aviv University, 2013 Measure and category 77 Proof. Given x 1 > x 2 >..., x k E n,r, x k x in ψ(r), we take t k ψ 1 (x k ) F n,r and note that t 1 > t 2 >..., t k t ψ 1 (x), f (n) (t k ) = r for all k, thus f (n) (t) = r, that is, t F n,r. If x is irrational then x E n,r since ψ 1 (x) = {t}. If x is rational then ψ 1 (x) = [s, t], and f (n) ( ) = r on [s, t] by Lemma 9a2 (applied to f (n) ). The case x k x is similar. 9a5 Exercise. Each E n,r is nowhere dense in ψ(r). Now Theorem 9a1 follows from the Baire category theorem (applied the second time). 9a6 Corollary. If an infinitely differentiable function f : R d R has only finitely many non-zero partial derivatives at every point then f is a polynomial. Proof. Let d = 2 (the general case is similar). By Theorem 9a1, for every x R the function f(x, ) : R R is a polynomial; similarly, each f(, y) is a polynomial. Introducing the set A n of all x R such that f(x, ) is a polynomial of degree n we have A n R, therefore A n is infinite (moreover, uncountable) for n large enough. The same holds for f(, y) and B n. For x A n the coefficients a 0 (x),..., a n (x) of the polynomial f(x, ) are linear functions of f(x, y 0 ),..., f(x, y n ) provided that y 0,..., y n B n are pairwise different. Therefore these coefficients are polynomials (in x), of degree n. We get a polynomial P : R 2 R such that f(x, y) = P (x, y) for x A n, y R. For every y R two polynomials f(, y) and P (, y) coincide on the infinite set A n, therefore they coincide on the whole R. A very similar (and a bit simpler) argument gives an interesting purely topological result. 9a7 Theorem. 1 If [0, 1] is the disjoint union of countably many closed sets then one of the sets is the whole [0, 1] (and others are empty). Proof. (sketch). Assume the contrary: [0, 1] = n F n, F n are closed. (Finitely many sets cannot do because of connectedness.) Then n Int F n is dense in [0, 1]. 1 Exercise 10:2.8 in Real analysis. Also Problem in book: B. Thomson, J. Bruckner, A. Bruckner, Elementary real analysis, second edition, 2008.

4 Tel Aviv University, 2013 Measure and category 78 Consider a (maybe empty) set P of all maximal nondegenerate intervals I [0, 1] such that n I F n. Note that intervals of P f are closed and pairwise disjoint. The open set G = I P Int I is dense in [0, 1], since it contains n Int F n. It follows that P, treated as a totally ordered set, is dense. Thus, the set C = [0, 1] \ G is perfect, with no interior (and in fact, homeomorphic to the Cantor set). As before, each F n C is nowhere dense in C. (Hint: if an endpoint of an interval I P belongs to F n C then I F n.) It remains to apply the Baire category theorem (in the second time). 9a8 Corollary. If the cube [0, 1] d is the disjoint union of countably many closed sets then one of the sets is the whole [0, 1] d (and others are empty). Proof. Let d = 2 (the general case is similar). Assume the contrary: [0, 1] 2 = n F n, F n are closed. By Theorem 9a7, each {x} [0, 1] is contained in a single F n. The same holds for each [0, 1] {y}. Thus, it is a single n. I wonder, is it true for an arbitrary continuum (that is, a compact connected metrizable space)? 9b Continuous and nowhere differentiable 9b1 Theorem. There exists a continuous function f : [0, 1] R such that for every x (0, 1), f is not differentiable at x. We consider the complete metric space C[0, 1] of all continuous f : [0, 1] R (separable, in fact). We define continuous functions ϕ n : C[0, 1] R by ϕ n (f) = ( k ( k 1 ) min f f. n) n k=1,...,n Clearly, ϕ n 0 pointwise. What about the rate of convergence? We take arbitrary ε n 0 and examine 1 ε n ϕ n. 1 9b2 Exercise. lim sup ϕ n (g) = for all f C[0, 1]. n,g f ε n By Prop. 5b9, (9b3) lim sup n 1 ε n ϕ n (f) =

5 Tel Aviv University, 2013 Measure and category 79 for quasi all f C[0, 1]. On the other hand, if f is differentiable at x 0 (0, 1) then f(x) f(x 0 ) = O( x x 0 ), that is, Taking k such that k 1 n Thus, C x [0, 1] f(x) f(x 0 ) C x x 0., k n [x 0 1 n, x n ] we get f( k n n ϕ n (f) 2C n. ) f ( k 1 n By (9b3), such f are a meager set, which proves Theorem 9b1. ) 2C n. 9b4 Exercise. There exists a continuous function f : [0, 1] R such that for every x (0, 1) lim sup f(y) f(x) log log log y x lim sup f(y) f(x) log log log y x+ 1 y x =, 1 y x =. However, f(y) f(x) cannot be replaced with f(y) f(x). If C > f(1) f(0) then there exists x (0, 1) such that lim sup y x+ f(y) f(x) y x C f(y) f(x) and moreover, sup y (x,1] C. Proof (sketch): choose b ( f(1) y x C, f(0) ) and take the greatest x such that f(x) Cx + b. 9c Differentiable and nowhere monotone 9c1 Theorem. 1 There exists a differentiable function f : [0, 1] R such that for every (a, b) [0, 1], f is not monotone on (a, b). 9c2 Lemma. 2 There exists a strictly increasing differentiable function f : [0, 1] R such that f ( ) = 0 on a dense set. 1 C.E. Weil (1976) On nowhere monotone functions, Proc. AMS 56, (Yes, two pages!) See also Sect in Real analysis. 2 S. Marcus (1963) Sur les dérivées dont les zéros forment un ensemble frontière partout dense, Rend. Circ. Mat. Palermo (2) 12, 5 40.

6 Tel Aviv University, 2013 Measure and category 80 Proof. We ll construct a continuous strictly increasing surjective g : [0, 1] [0, 1] such that the inverse function f = g 1 : [0, 1] [0, 1] has the needed properties. It is sufficient to ensure that (finite or infinite) derivative g ( ) (0, ] exists everywhere (and never vanishes), and is infinite on a dense set. A function α(x) = x 1/3 is strictly increasing (on R), with α (0) = + and α (x) (0, ) for x 0. We introduce α(1 + h) α(1) A = max (0, ) h 0 hα (1) (this continuous function vanishes on ± ; in fact, A = 4) and note that (9c3) α(x + h) α(x) hα (x) A for all h 0 and x (since for x 0 it equals x1/3 (α(1+ h x ) α(1)) hx 2/3 α (1) = α(1+ h x ) α(1) h x α (1) Similarly to Sect. 5a we choose some a n, c n (0, 1) such that a n are pairwise distinct, dense, and n c n <. The series β(x) = c n α(x a n ) n=1 converges uniformly on [0, 1] (since α( ) 1 and n c n < ). The series n=1 c nα (x a n ) converges (to a finite sum) for some x and diverges (to + ) for other x (in particular, for x {a 1, a 2,... }). We consider β n (x) = n k=1 c kα(x a k ) and γ n (x) = β(x) β n (x) = k=n+1 c kα(x a k ). By (9c3), 0 γ n(x + h) γ n (x) h A k=n+1 for all h 0 and x. Thus (similarly to Sect. 5a) β n(x) β(x + h) β(x) lim inf } {{ } h 0 h n k=1 c kα (x a k ) therefore lim sup h 0 β (x) = β(x + h) β(x) h c k α (x a k ) β n(x) + A c n α (x a n ) (0, ] n=1 k=n+1 c k α (x a k ), ).

7 Tel Aviv University, 2013 Measure and category 81 for all x. It remains to take g(x) = β(x) β(0) β(1) β(0). Do not think that β ( ) = only on the countable set {a 1, a 2,... }. Amazingly, f (x) = 0 for quasi all x [0, 1] (and therefore β (x) = for quasi all x [0, 1]). Here is why. By 5b2 and 5c5, f is of Baire class 1, thus, {x : f (x) 0} is an F σ set, and {x : f (x) = 0} is a G δ set; 1 being dense it must be comeager (as noted before 5c2). We introduce the space D of all bounded derivatives on (0, 1); that is, of F for all differentiable F : (0, 1) R such that F is bounded. We endow D with the metric ρ(f, g) = sup f(x) g(x). x (0,1) 9c4 Exercise. (a) D is a complete metric space. (b) D is not separable. We consider a subspace D 0 of all f D such that f(x) = 0 for quasi all x. As noted above, this happens if and only if f( ) = 0 on a dense set. By 9c2, D 0 is not {0}; moreover, for every x (0, 1) there exists f D 0 such that f(x) 0 (try f(ax + b)). 9c5 Exercise. (a) D 0 is a vector space; that is, a linear combination of two functions of D 0 is a function of D 0. (b) D 0 is a closed subset of D. Given (a, b) (0, 1), the set E a,b = {f D 0 : x (a, b) f(x) 0} is closed (evidently). Given f E a,b, we take x (a, b) such that f(x) = 0 and g D 0 such that g(x) > 0. Then f εg D 0 and f εg / E a,b for all ε > 0; thus, f is not an interior point of E a,b. We see that E a,b is nowhere dense. Similarly, E a,b = {f D 0 : x (a, b) f(x) 0} is nowhere dense. It follows that quasi all functions of D 0 change the sign on every interval. Theorem 9c1 is thus proved. 1 A straightforward representation f (x) = 0 ε δ h ( h < δ = f(x + h) f(x) ε h ) gives only F σδ. Taking into account that f is differentiable we have another representation f (x) = 0 ε h ( h < ε f(x + h) f(x) < ε h ) that gives G δ.

8 Tel Aviv University, 2013 Measure and category 82 Hints to exercises 9a5: otherwise, some interval of P f is not maximal. 9b2: g( k n ) = f( k n ) ± ε n. 9b4: similar to 9b1. 9c4: (a) D is closed in the space of all bounded functions; (b) try shifts of a discontinuous derivative.

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