Lipschitz_Function

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Definition

A function $f:A \rightarrow \mathbb{R}$ is called Lipschitz if there exists a bound $M > 0$ such that

$$ \left| \frac{f(x) - f(y)}{x - y} \right| \leq M $$

for all $x \neq y \in A$.

Equivalently, $|f(x) - f(y)| \leq M|x - y|$ for all $x, y \in A$.

Geometrically, a function $f$ is Lipschitz if there is a uniform bound on the magnitude of the slopes of lines drawn through any two points on the graph of $f$.

Proof (Lipschitz implies Uniform Continuity)

If $f$ is Lipschitz with constant $M$, then given $\epsilon > 0$, choose $\delta = \epsilon/M$.

For $|x - y| < \delta$:

$$ |f(x) - f(y)| \leq M|x - y| < M \cdot \frac{\epsilon}{M} = \epsilon $$

Thus $f$ is uniformly continuous.