Exercise of comager set and the space $C([0,1])$

Question

Let $X$ be a topological space and $P \subseteq X$ holds generic. If $P$ is comeager , we say that $P$ hold generically or that the generic element of $X$ is in $P$

Show that the generic element of $C([0,1])$ is nowhere differentiable ( so there exist differentiable nowhere functions.)

This exercise is from book Classical Descriptive Set Theory Kechris.

A suggestion for this exercise please.

Answer 1

Prove

$(1)$ Prove that given $m \in \mathbb{N}$, any function $f \in{} C([0; 1])$ can be approximated (in the uniform metric) by a piecewise linear function $g \in{}C([0; 1])$, whose linear pieces (finitely many) have slope $±M$, for some $M \geq m$.

$(2)$ For each $n \geq 1$, let $E_n$ be the set of all functions $f \in{} C([0; 1])$, for which there is $x_0 \in{} [0; 1]$ (depending on $f$) such that $|f(x) − f(x_0)|\leq n|x − x_0|$ for all $x \in{} [0; 1]$. Show that $E_n$ is nowhere dense using the fact that if $g$ is as in $(1)$ with $m = 2n$, then some open neighborhood of $g$ is disjoint from $E_n$.