Uniformly continuous and differentiable. 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.

1)
Let $ε>0$. Choose $δ=?$. Exercises like these are about choosing a right $δ>0$.
We know that there exist a number $K≥0$, such that $|f(x)-f(y)|≤K|x-y|$. (1)
Note that: $|x-y|<δ$ implies that $(K+1)|x-y|<(K+1)δ$. (as $K+1>0$). (2)
Combining (1) and (2) gives: 
$$|f(x)-f(y)|≤K|x-y|<(K+1)|x-y|<(K+1)δ$$
If we manage to get $(K+1)δ$ equal to $ε$, than we are done.
We can't choose $ε$ tough, but we can choose $δ$.
Therefore we choose $δ=\frac{ε}{K+1}$.
Note that $δ>0$ because $K+1>0$ and $ε>0$.
