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 $\map {\mathcal D} I$ 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 form 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:


 * $\map {\mathcal D} I$ is meager in $\struct {\map \CC I, d}$.

So:


 * $\map {\mathcal D} I \ne \map \CC I$.

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

Examples
An explicit example is given in Weierstrass's Theorem.