Let $D\subseteq \mathbb{R}$ and let $f:D\rightarrow \mathbb{R}$. We say that the function $f$ is an $\mathcal{L}$-function if there exists some constant $K\geq 0$ for which $\left|f(x)-f(y)\right|\leq K\left|x-y\right|$.
1.) Prove that every $\mathcal{L}$-function function is uniformly continuous on its domain.
2.) Give an example showing that there exist uniformly continuous functions which are not $\mathcal{L}$-functions.
3.) Prove that if $f:(a,b)\rightarrow \mathbb{R}$ is an $\mathcal{L}$-function and is differentiable, then $f'$ is bounded.
4.) Prove or disprove that a function is an $\mathcal{L}$-function on $(a,b)$ if and only if it is differentiable on $(a,b)$.
Response: So far I haven't gotten any work worth showing.