Continuous Real Function Differentiable on Borel Set

Theorem
Let $\map \BB {\R, \size {\, \cdot \,} }$ be the Borel Sigma-Algebra on $\R$ with the usual topology.

Let $f: \R \to \R$ be a continuous real function.

Let $\map D f$ be the set of all points at which $f$ is differentiable.

Then $\map D f$ is a Borel Set with respect to $\map \BB {\R, \size {\, \cdot \,} }$.

Proof
By the definition of derivative:


 * $\map {f'} x$ exists




 * $\displaystyle \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h = L$ exists

where $L = \map {f'} x$.

For ease of presentation, denote:


 * $\Delta f := \map f {x + h} - \map f x$

Since $f$ is continuous by hypothesis, so too is $\dfrac {\Delta f} h$ for $h \ne 0$.

Thus we will prove an inequality for $\dfrac {\Delta f} h$ for all rational $h$.

The inequality will then extend to all real $h$ by Rationals are Everywhere Dense in Topological Space of Reals.

By the definition of limit, the claim that the derivative exists is:


 * $\exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall h \in \R \setminus \set 0: \size h < \delta \implies \size {\dfrac {\Delta f} h - L} < \epsilon$

From Limit with Rational $\epsilon$ and $\delta$ and Limit with $\epsilon$ Powers of $2$, we can instead consider:


 * $\exists L \in \R: \forall n \in \N: \exists \delta \in \Q_{>0}: \forall h \in \Q \setminus \set 0: \size h < \delta \implies \size {\dfrac {\Delta f} h - L} < 2^{-n}$

Let $\set {\delta_n}$ be an enumeration of all such $\delta$s considered.

Assume the following:


 * $\forall n \in \N: \exists \delta_n \in \Q_{>0}: \exists y_n \in \Q: \forall h \in \closedint {-\delta_n} {\delta_n}: \size {\map f {x + h} - \map f x - h y_n} \le \size h 2^{-n}$

for some rational sequence $\sequence {y_n}_{n \mathop \in \N}$.

Then:

By the Comparison Test, $\displaystyle \sum_{n \mathop \ge 0} \paren {y_{n + 1} - y_n}$ converges.

The difference of successive terms makes this a Telescoping Series:


 * $\displaystyle \sum_{n \mathop \ge 0} \paren {y_{n + 1} - y_n} = -y_0 + \lim_{n \mathop \to \infty} y_n$

so $y_n$ converges to a limit, which will be called $y$.

Then:

We have thus shown that the assumption that:


 * $\forall n \in \N: \exists \delta_n \in \Q_{>0}: \exists y_n \in \Q: \forall h \in \closedint {-\delta_n} {\delta_n}: \size{ \map f {x + h} -\map f x - h y_n } \le \size h 2^{-n}$

... implies that for all $n \in \N, -\delta_n < h < \delta_n$, there is a real number $y$ such that:


 * $\size {\Delta f - h y} \le \size h \paren {2^{-n} + \size {y_n - y} }$

Equivalently:


 * $\size {\dfrac {\map f {x + h} - \map f x} h - y} \le \paren {2^{-n} + \size {y_n -y} }$

The tends to $0$ as $n \to \infty$, as $2^{-n} \to 0$ and $y_n \to y$.

Thus the also tends to $0$ as $n \to \infty$.

Thus $\dfrac{\map f {x + h} - \map f x} h \to y$ as $n \to \infty$.

Then $y = \map {f'} x$ by the definition of derivative, and we have shown that our assumptions imply the existence of $\map {f'} x$.

The set of all $x \in \R$ satisfying:


 * $\forall n \in \N: \exists \delta_n \in \Q_{>0}: \exists y_n \in \Q: \forall h \in \closedint {-\delta_n} {\delta_n}: \size {\map f {x + h} - \map f x - h y_n} \le \size h 2^{-n}$

is:


 * $\displaystyle \bigcap_n \bigcup_{\delta_n} \bigcup_{y_n} \bigcap_h \set {x \in \R: \size {\dfrac {\map f {x + h} - \map f x} h - y_n} < 2^{-n} }$

By hypothesis, $f$ is continuous.

Hence each set so indexed is open.

So it is seen that $\map D f$ has been constructed from countably many intersections and unions of open sets.

Hence $\map D f$ is a Borel Set by definition.