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In page Lipschitz continuity:

"
  • Conversely, if f : I → R is absolutely continuous and thus differentiable almost everywhere, and satisfies |f′(x)| ≤ K for almost all x in I, then f is Lipschitz continuous with Lipschitz constant at most K.
  • More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map f : U → Rm, where U is an open set in Rn, is almost everywhere differentiable. Moreover, if K is the best Lipschitz constant of f, then ‖ D f ( x ) ‖ ≤ K {\displaystyle \|Df(x)\|\leq K} whenever the total derivative Df exists.[citation needed]