The exercise 14.13 of the Bass' Analysis Real for Graduate Students asks for an increasing absolutely continuous function from $[0,1]$ to $\mathbb{R},$ s.t. $\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}=\infty$ in $A$, given $A\in[0,1]$ of Lebesgue measure zero.
The hint is to choose open sets $G_n$ containing $A$ such that $m(G_n) < 2^{−n}$, let $h_n(x) = \int_0^x\chi_{G_n}(y) dy$ and $f=\sum h_n$.
My intuiton is really challenged with this. My thought is that $\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}=\infty$ if I have a vertical assimptotyc. How can I have a continuous function? What is the graphic?
For instance, $A=\{1/2\}$. How can I have $\lim_{h\to 0}\dfrac{f(1/2+h)-f(1/2)}{h}=\infty$ and the function be continuous?
Thank you in advance.
This is a particularly useful result that is usually credited to the French mathematician Gustave Choquet [1] from 1947. Independently the Polish mathematician Zygmund Zahorski [2] proved it a few years later.
I wouldn't have wanted to be assigned this myself as a graduate student.
The theorem and its proof is reproduced in Andy Bruckner's monograph [3].
Theorem. [Choquet] Let $Z\subset [0,1]$ be a set of type $\cal G_\delta$ and of measure zero. There exists an absolutely continuous function $G$ defined on $[0,1]$ so that $G'(x)=\infty$ for all $z\in Z$ and $\infty> G'(x)\geq 1$ for all $x\in [0,1]\setminus Z$.
In particular $G$ is strictly increasing and is the indefinite integral of its derivative.
If you need orientiation on this, Andy's monograph is of considerable assistance. If you are just lurking here then it is well that you at least know the history and the fact that this is not some make-work exercise but a useful tool in the study of derivatives. If it has never occurred to you that there is a definition and use for infinite derivatives, well now you know.
As an interesting application of this theorem you can deduce this one:
Theorem [Bruckner-Goffman [4]]: Suppose that $f$ is a real-valued function defined on the closed interval $[0,1]=I$. There is a homeomorphism $h$ of $I$ onto $I$ such that $f∘h$ is differentiable on $I$ and $(f∘h)′$ is bounded on $I$ if and only if $f$ is continuous and of bounded variation.
References:
[1] Choquet, Gustave, Application des propriétés descriptives de la fonction contingent à la théorie des fonctions de variable réelle et à la géométrie différentielle des variétés cartésiennes. J. Math. Pures Appl. (9) 26 (1947), 115–226 (1948).
[2] See Theorem 8 in Z. Zahorski, Sur la premiere derivee, Trans. Amer. Math. Soc. 69 (1950), 1-54.
[3] A proof appears on page 86 in this monograph: Bruckner, Andrew. Differentiation of real functions. Second edition. CRM Monograph Series, 5. American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6
[4] Bruckner, A. M.; Goffman, C. Differentiability through change of variables. Proc. Amer. Math. Soc. 61 (1976), no. 2, 235–241 (1977).