Lipschitz Continuity

analysis

Definition

Lipschitz Continuity

A function is Lipschitz continuous on if one constant bounds all output changes by the corresponding input changes. Formally, there exists such that, for all ,

Any such is called a Lipschitz constant. The inequality says that no secant line of the graph can have absolute slope larger than . The function may bend, but it may not become arbitrarily steep.

Uniform control

Lipschitz continuity is a global version of bounded steepness. The same constant must work for every pair of points in the domain. Rearranging the defining inequality gives

so every secant slope is controlled at once.

This immediately implies uniform continuity. Given , choose when . Then forces . If , the function is constant and the claim is trivial.

Relation to derivatives

On an interval, a differentiable function with bounded derivative is Lipschitz continuous. If

then the mean value theorem gives

Conversely, if a Lipschitz function is differentiable at a point, then the derivative at that point has absolute value at most its Lipschitz constant. A Lipschitz function need not be differentiable everywhere, but Rademacher’s theorem says that Lipschitz functions on Euclidean space are differentiable almost everywhere.

Examples and non-examples

• Every constant function is Lipschitz continuous with .
• The function is Lipschitz continuous on with .
• The function is not Lipschitz continuous on all of , because its slope grows without bound. It is Lipschitz continuous on every bounded interval.
• The function is uniformly continuous on , but it is not Lipschitz continuous there because its slope becomes unbounded near .