Derivative of Differentiable Function on Open Interval is Somewhere Continuous

Theorem

Let $I = \openint a b$ be a open interval..

Let $f : I \to \R$ be a differentiable function.

Let $f'$ be the derivative of $f$.

Then $f'$ has a point of continuity.

Proof

Let $D$ be the set of discontinuities of $f'$.

Let $d$ be Euclidean metric on $\R$.

From Derivative of Differentiable Function on Open Interval is Baire Function:

$f'$ is a Baire function.

From Set of Discontinuities of Baire Function is Meager:

$D$ is meager in $\struct {I, d}$.

From Open Real Interval is Non-Meager:

$I$ is non-meager in $\struct {I, d}$.

Therefore $D \ne I$.

Since $D \subset I$, we have that $I \setminus D$ is non-empty.

That is, $f'$ has a point of continuity.

$\blacksquare$