Question: Suppose $f$ is differentiable in an interval $E \subset R$. Prove that $f'$ is bounded on $E$ if and only if a constant $M$ exists such that $|f(x) − f(y)| \le M|x − y|$ for all $x, y \in E$.
My approach: I'll skip the part about the $\Leftarrow$ direction since it is quite straightforward.
Assume that $f'$ is bounded.
Set $\phi(t)$ as $f(t)−f(x)/t−x$. Then
$$|\lim_{t\to x} \phi(t)| = |f'(x)| \le M$$
according to the characteristic of boundedness.
The next step is the part where I'm unsure whether it is legit.
multiply $|t-x|$ on both sides of inequality then it will derive $|f(t) − f(x)| \le M|t − x|$.
Is this approach appropriate? Or is there more legitimate way to prove this condition?
Try to apply Mean Value Theorem, keeping in mind that $|f’(x)|<M$ for some $M>0$.