Set of nowhere differentiable function in C[(0,1]) is dense

Question

How can we prove with Baire's Theorem that in C[(0,1]), the set of nowhere differentiable function is dense.

Answer 1

Here is a sketch:

$1).$ Show that $C([0,1])$ is not the countable union of nowhere dense sets.

$2).$ Set

$D=\{f\in C([0,1]): f \text{ is differentiable at some $x$ \}}$

and

$A_{n,m}=\left \{ f\in C([0,1]): \frac{f(t)-f(x)}{t-x})<n \ \text{if}\ 0<|x-t|<\frac{1}{m}\right \}$.

$3).\ D\subseteq A_{n,m}$ and $A_{n,m}$ is closed.

$4).\ A_{n,m}$ are nowhere dense. (This is the only really hard part).

$5).$ Conclude.