Suppose $f$ is differentiable and not uniformly continuous show that $|f'|$ is not bounded by $R$

Question

Show for any constant $C$, $|f'(t)|> C$ for some $t$ exits in $R$.

I can see why this is true but just cannot write up a good solution.

Answer 1

Hint: Suppose that $f'$ is bounded by $M$ $x\leq y$ MVT implies that $\mid f(x)-f(y)\mid\leq \mid f'(c)\mid \mid x-y\mid\leq M\mid x-y\mid$ where $c\in [x,y]$. For every $\epsilon>0$ and $x,y$ such that $\mid x-y\mid <{\epsilon\over M}$, $\mid f(x)-f(y)\mid \leq M{\epsilon\over M}<\epsilon$