Lipschitz continuous function
High school
Consider a function $f$ and the lines through pairs of points on its graph. The function $f$ is Lipschitz continuous if there is a number $L$ such that every one of those lines has a slope between $-L$ and $L$.
Undergraduate (metric spaces)
Consider a function $f : X \to Y$ between metric spaces. The function $f$ is Lipschitz continuous if there exists $L \geq 0$ such that
\[d(f(x), f(y)) \leq L d(x, y) \quad \text{for all } x, y \in X.\]The smallest such $L$ is the Lipschitz constant of $f$.
Examples
Examples
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$x \mapsto \abs{x}$ on $[-1, 1]$
High school
$x \mapsto \abs{x}$ is Lipschitz continuous with constant $1$, because $\abs{\abs{x} - \abs{y}} \leq \abs{x - y}$.
Non-examples
Things that may be mistaken for examples.
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$f(x) = x^2$ on $\reals$
High school
$x \mapsto x^2$ on $\reals$ is not Lipschitz continuous, because its slope $2x$ is unbounded on $\reals$.
Properties
- Every Lipschitz continuous function is uniformly continuous.
- A Lipschitz continuous function on an interval is absolutely continuous.
- A Lipschitz continuous function on $\reals^n$ is differentiable almost everywhere (Rademacher’s theorem).
- A differentiable function on an interval is Lipschitz continuous with constant $L$ if and only if $\abs{f^{\prime}(x)} \leq L$ for all $x$ (by the mean value theorem).
- If $f : \reals^n \to \reals^n$ is Lipschitz continuous, then for every initial condition $x_0 \in \reals^n$, the differential equation $\dot{x} = f(x)$ with $x(0) = x_0$ has exactly one solution, and the solution is defined for all $t \in \reals$ (the global Picard–Lindelöf theorem). If $f$ is only locally Lipschitz continuous, then the solution exists and is unique on some time interval around $0$, but the solution can blow up in finite time: the solution of $\dot{x} = x^2$ with $x(0) = 1$ is $x(t) = 1 / (1 - t)$, which blows up at $t = 1$.