Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Corollary
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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.
- $\map \CC I$ is non-meager in $\struct {\map \CC I, d}$.
- $\map \DD I$ 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.