Theorems · Theorem · general topology

LocallyLipschitz.continuous

∀ {α : Type u} {β : Type v} [inst : PseudoEMetricSpace α] [inst_1 : PseudoEMetricSpace β] {f : α → β},
  LocallyLipschitz f → Continuous f

A locally Lipschitz function is continuous. (The converse is false: for example, $x ↦ \sqrt{x}$ is continuous, but not locally Lipschitz at 0.)

Defined in

Mathlib.Topology.EMetricSpace.Lipschitz

Cited by

2 results in Mathlib

Foundations

Depth 152 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoEMetricSpacePseudoEMetricSpace