Lipschitz Continuity and Bounded Derivatives of Differentiable Functions
Let $I$ be an interval of $\mathbb{R}$ and $f:I\to\mathbb{R}$ be differentiable.
Show that $f$ is a Lipshitz continuous function if, and only if, its derivative is bounded.
Show Answer:
$(\Leftarrow)$ In accordance with the mean value inequality.
$(\Rightarrow)$ If $f$ is $k$ Lipshitz continuous then for all $x,y\in I$ such that $x\ne y$ we have
$$\left|\frac{f(x)-f(y)}{x-y}\right|\le k.$$
In the limit as $y\to x$, we obtain $∣f'(x)∣\le k$. Consequently, $f'$ is bounded.