How do I prove that the set
$L_n = \{f \in C([0, 1]), \exists \, x_0 \text{ such that } \forall \, x \in [0, 1], |f(x) - f(x_0)| \leq n|x - x_0|\}$
is closed with empty interior? Furthermore, how do I show that $C([0, 1]) \setminus \cup_n L_n$ is of second category and it is made of nowhere differentiable functions.
This is my idea:
The set $L_n = \{f \in C([0, 1]) \mid \exists \, x_0 \in [0, 1], \forall \, x \in [0, 1], |f(x) - f(x_0)| \leq n|x - x_0|\}$ is closed because it is defined by a Lipschitz condition, and uniform limits of functions satisfying this condition also satisfy it. Specifically, if a sequence ({f_k} \subset L_n) converges uniformly to $f$, the associated points $\{x_k\}$ converge (up to a subsequence) to some $x_0$, and the uniform convergence ensures that $|f(x) - f(x_0)| \leq n|x - x_0|$, proving $f \in L_n$. How do I conclude the rest?
I would appreciate any helpful suggestions, ideas, or comments.