Lipschitz continuity

Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function, and let $K \ge 0$ be real. The function $f$ is Lipschitz with constant $K$ if $$|f(x) - f(y)| \le K |x - y| \quad \text{for all } x, y \in D.$$ It is Lipschitz if it is Lipschitz with some constant $K$.

Remarks

Lipschitz functions are uniformly continuous. A differentiable function on an interval with $|f'| \le K$ is Lipschitz with constant $K$. For example, $|x|$ and $\sin x$ are Lipschitz with constant $1$. The function $\sqrt{x}$ on $[0, 1]$ is not Lipschitz. A Lipschitz map with constant $K < 1$ from a set to itself is a contraction. Mathlib defines it as LipschitzWith.

Explanations

L1 human

The slope of each chord of the graph of a Lipschitz function is at most $K$ in absolute value. So the function never changes faster than a fixed rate.