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

Theorem
Let $I = \closedint a b$.

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

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.

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

Proof
Let:


 * $\ds A_{n, \, m} = \set {f \in \map \CC I: \text {there exists } x \in I \text { such that } \size {\frac {\map f t - \map f x} {t - x} } \le n \text { for all } t \text { with } 0 < \size {t - x} < \frac 1 m}$

and:


 * $\displaystyle A = \bigcup_{\tuple {n, \, m} \in \N^2} A_{n, \, m}$

Lemma 2
Since each $\struct {A_{n, \, m}, d}$ is nowhere dense in $\struct {\map \CC I, d}$:


 * $A$ is the union of nowhere dense sets.

So, by the definition of a meager space:


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

By Subspace of Meager Space is Meager Space and Lemma 1, we have:


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