Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Corollary

Theorem

Let $I = \closedint a b$.

Let $\map \CC I$ be the set of continuous functions on $I$.

Then:

there exists a function $f \in \map \CC I$ that is not differentiable anywhere.

Proof

Let $\map \DD I$ be the set of continuous functions on $I$ that are differentiable at a point.

Let $d$ be the metric induced by the supremum norm.

By Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Banach Space:

$\struct {\map \CC I, d}$ is a complete metric space.

By Baire Space is Non-Meager:

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

By Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval:

$\map \DD I$ is meager in $\struct {\map \CC I, d}$.

So:

$\map \DD I \ne \map \CC I$.

That is, there exists a continuous function that is not differentiable anywhere.

$\blacksquare$

Examples

An explicit example is given in Weierstrass's Theorem.