Locally Lipschitz continuous function
High school
Consider a function $f$ and the lines through pairs of points on its graph. The function $f$ is locally Lipschitz continuous if each point has a surrounding interval in which the slopes of those lines are bounded, though the bound can be different for different intervals.
Undergraduate (metric spaces)
Consider a function $f : X \to Y$ between metric spaces. The function $f$ is locally Lipschitz continuous if every point $x_0 \in X$ has a neighborhood $U$ and a number $L_U \geq 0$ such that
\[d(f(x), f(y)) \leq L_U \, d(x, y) \quad \text{for all } x, y \in U.\]Examples
Examples
-
$f(x) = x^2$ on $\reals$
High school
$x \mapsto x^2$ on $\reals$ is locally Lipschitz continuous, because on each interval $[-R, R]$ its slope $2x$ is bounded by $2R$.
-
$x \mapsto \abs{x}$ on $[-1, 1]$
High school
$x \mapsto \abs{x}$ on $[-1, 1]$ is locally Lipschitz continuous, because the function is Lipschitz continuous on all of $[-1, 1]$ with constant $1$, even at the corner at $0$.
Properties
- Every continuously differentiable function on an open subset of $\reals^n$ is locally Lipschitz continuous.
- A locally Lipschitz continuous function is (globally) Lipschitz continuous on every compact subset of its domain.
- If $f$ is locally Lipschitz continuous, then the differential equation $\dot{x} = f(x)$ has a unique solution from each initial condition on some time interval, but the solution can blow up in finite time, as for $\dot{x} = x^2$. If $f$ is Lipschitz continuous on $\reals^n$, then each solution exists for all time.