Continuous functions are differentiable on a measurable set?

Here's an attempt to salvage Matthew Pancia's solution, which unfortunately depended on an uncountable union over all possible derivatives.

Given $f$ we can define, in the obvious way, a continuous function $F:\mathbb R\times(\mathbb R\setminus 0)\to \mathbb R$ such that $f$ is differentiable at $x$ exactly when $\lim_{h\to 0} F(x,h)$ exists. The usual formalization of this is $$\exists y:\forall\varepsilon:\exists \delta:\forall h: |h|<\delta\Rightarrow |F(x,h)-y|<\varepsilon$$ Classically all of the variables here are real, but it is easy to see that we can restrict $\varepsilon$ and $\delta$ to $\mathbb Q$ without changing the meaning. We can also restrict $h$ to $\mathbb Q$ because $F$ is continuous. However, it is essential that $y$ can be an arbitrary real, because otherwise we would be looking for points where $f$ is differentiable with rational derivative, which is something quite different.

However, we can also formalize the existence of a limit like $$\forall\varepsilon:\exists \delta:\exists Y:\forall h: |h|<\delta \Rightarrow |F(x,h)-Y|<\varepsilon$$ This works because $\mathbb R$ is complete; it is essentially the same as changing "has a limit" about a sequence to "is Cauchy". The arguments that $\varepsilon$, $\delta$ and $h$ can be restricted to the rationals work as before, but now $Y$ can also be taken to be a rational in each case.

For each particular choice of $\varepsilon$, $\delta$, $Y$, and $h$, the set of $x$ such that $|h|<\delta\Rightarrow |F(x,h)-Y|<\varepsilon$ is open and therefore Borel.

Now handle each of the quantifiers from the inside out: For each choice of $\varepsilon$, $\delta$, and $Y$, the set of $x$ such that $$\forall h: |h|<\delta \Rightarrow |F(x,h)-Y|<\varepsilon$$ is a countable intersection of Borel sets and therefore Borel. For each choice of $\varepsilon$ and $\delta$ the set of $x$ such that $$\exists Y:\forall h: |h|<\delta \Rightarrow |F(x,h)-Y|<\varepsilon$$ is a countable union of Borel sets and therefore Borel. And so forth. At the top we find that the set of points of differentiability is Borel and thus in particular measurable.

Theorem 1 Let $f$ be a measurable function on $(a,b)$. Then the function $$ g_n(x)=\sup\left\{f(x+h):h\in(0,\frac{1}{n})\cap(0,b-x)\right\} $$ is measurable.

Proof Fix $c\in\mathbb{R}$. We want to prove that the set $A=\{x\in(a,b):g_n(x)>c\}$ is measurable, then $g_n$ will be measurable. Let $x_0\in A$, then there exist $h_0\in(0,\frac{1}{n})\cap(0,b-x_0)$ such that $f(x_0+h_0)>c$. Define $$ \delta_1=\frac{1}{2}\left(\min\left(\frac{1}{n},b-x_0\right)-h_0\right)\qquad \delta_2=\frac{1}{2}\left(\min\left(\frac{1}{n},b-x_0\right)+h_0\right) $$ then for all $x\in(x_0-\delta_1, x_0+\delta_2)$ we have $x_0+h_0\in\{x+h:h\in(0,\frac{1}{n})\cap(0,b-x)\}$. As the consequence $g_n(x)\geq f(x_0+h_0)>c$, so $x\in A$ for all $x\in(x_0-\delta_1, x_0+\delta_2)$. This means that $A$ is open, hence measurable.

Theorem 2 Let $f$ be a mesurable function on $(a,b)$ then the set $$ A=\{x\in(a,b):\exists f'(x)\in\mathbb{R}\} $$ is measurable.

Proof Extend $f$ by equalities: $f(x)=f(a)$ for $x<a$ and $f(x)=f(b)$ for $x>b$. For each $c\in\mathbb{R}$ consider measurable function $\phi(x)=f(x)-cx$. From theorem 1 it follows that $$ \psi_n(x)=\sup\left\{\phi(x+h):h\in(0,\frac{1}{n})\cap(0,b-x)\right\} $$ is measurable. Then we have measurable function $$ \psi_n(x)-\phi(x)=\sup\left\{\phi(x+h)-\phi(x):h\in(0,\frac{1}{n})\cap(0,b-x)\right\}. $$ Hence the set $$ B=\{x\in(a,b):\psi_n(x)-\phi(x)>0\}= $$ $$ \left\{x\in(a,b):\sup\left\{\frac{f(x+h)-f(x)}{h}:h\in(0,\frac{1}{n})\cap(0,b-x)\right\}>c\right\} $$ is measurable. Since $c\in\mathbb{R}$ is arbitrary then the functions $$ f_n(x)=\sup\left\{\frac{f(x+h)-f(x)}{h}:h\in(0,\frac{1}{n})\cap(0,b-x)\right\} $$ are measurable. So we conclude that the function $$ \overline{f}'_+(x)=\overline{\lim\limits_{h\to 0+}}\frac{f(x+h)-f(x)}{h}=\lim\limits_{n\to\infty}f_n(x) $$ is also measurable. Similarly, we can prove that functions $$ \underline{f}'_+(x)=\underline{\lim\limits_{h\to 0+}}\frac{f(x+h)-f(x)}{h}=\lim\limits_{n\to\infty}f_n(x) $$ $$ \overline{f}'_-(x)=\overline{\lim\limits_{h\to 0-}}\frac{f(x+h)-f(x)}{h}=\lim\limits_{n\to\infty}f_n(x) $$ $$ \underline{f}'_-(x)=\underline{\lim\limits_{h\to 0-}}\frac{f(x+h)-f(x)}{h}=\lim\limits_{n\to\infty}f_n(x) $$ are mesurable. Finally, the set $A=\{x\in(a,b):\exists f'(x)\in\mathbb{R}\}$ is measurable because $$ A=\{x\in(a,b):\overline{f}'_+(x)=\underline{f}'_+(x)=\overline{f}'_-(x)=\underline{f}'_-(x)\in\mathbb{R}\} $$

What follows is taken from a sci.math post of mine from 16 May 2006.

Let $f:{\mathbb R} \rightarrow {\mathbb R}$ be arbitrary. Then the set of points where $f$ does not have a finite derivative is a $G_{\delta \sigma}$ set. This is also true for the set of points where $f$ does not have a finite left derivative or the set of points where $f$ does not have a finite right derivative. A proof of this is given after the next paragraph.

Conversely, given any $G_{\delta}$ set $G_1$ and $G_{\delta \sigma}$ set $G_2$ such that $G_2$ has Lebesgue measure zero, then there exists a continuous function $f:{\mathbb R} \rightarrow {\mathbb R}$ whose non-differentiability set is precisely $G_1 \cup G_2.$ This was proved by Zygmunt Zahorski (1914-1998) in 1941 and re-published in French in 1946 (Zbl 61.11302; MR 9,231a). For the 1946 French paper, see http://tinyurl.com/3zwca22.

The first statement above for continuous $f$ follows from the fact that $f$ has a finite derivative at $x$ if and only if $x$ belongs to

$$\bigcap_{k=1}^{\infty} \;\bigcup_{n=1}^{\infty} \;\; \bigcap_{0 < |\eta| < \frac{1}{n}} \;\; \bigcap_{0 < |\delta| < \frac{1}{n}} \; \left\{x:\;\; \left| \frac{f(x + \eta) - f(x)}{\eta} \;\; - \;\; \frac{f(x + \delta) - f(x)}{\delta} \right| \; \leq \frac{1}{k}\right\}$$

To see that this result holds for an arbitrary function $f$ (not necessarily continuous), we relativize the above decomposition to $C(f)$, the continuity set of $f$. That is, consider the sets $\{x \in C(f): |$ stuff $| \leq \frac{1}{k}\}.$ This will show that the set where $f$ has a finite derivative is an $F_{\sigma \delta}$ subset of the space $C(f),$ where $C(f)$ has the subspace topology from ${\mathbb R}.$ ("Subset", because every point at which a finite derivative exists is a point of continuity.) Hence, this set is the intersection of $C(f)$ with an $F_{\sigma \delta}$ set in ${\mathbb R}$ (subspace topology stuff), and therefore this set is $F_{\sigma \delta}$ in ${\mathbb R}$ (because $C(f)$ has a lower Borel class, being $G_{\delta}$ in ${\mathbb R}).$