Let $M_K$ be the set of all function $f$ in $C_{[a,b]}$ satisfying a Lipschitz condition, i.e., the set of all $f$ such that $$|f(t_1)-f(t_2)|\leq K|t_1-t_2|$$ for all $t_1,t_2 \in [a,b]$ where $K$ is a fixed positive number. Prove that:
a) $M_K$ is closed and in fact the closure of the set of all differentiable functions on $[a,b]$ such that $|f'(t)| \leq K$
b)The set $M= \cup_k M_k$ of all functions satisfying a Libschtiz condition for some $K$ is not closed.
c)The closure of $M$ is the whole space $C_{[a,b]}$
My attempt
a) Let $M_k=\{f:[a,b] \rightarrow \mathbb{R}: |f(x)-f(y)| \leq K|x-y| , \forall x,y$
Let $D_k=\{f:[a,b]\rightarrow \mathbb{R}:\text{f differentiable}\\|f'(x)|\leq K, \forall x\in[a,b] \}$
The goal is to show that $\bar{D_k} \subseteq M_k$, and $M_k \subseteq \bar{D_k}$. We can do so by showing that $M_k$ is closed by showing that it contains all its limit points and then show that any cauchy sequence in $D_k$ converges in $M_k$. This would show that the closure of $D_k$ is in of $M_k$
b) Not sure
c) We want to show that $M \subset C_{[a,b]}$ and $C_{[a,b]} \subset M$
for the $\Leftarrow$ direction I think we can show that for any cauchy sequence in $M$ the limit is continuous.