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.

(\Leftarrow)

In accordance with the mean value inequality.

(\Rightarrow)

f

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.