Lipschitz condition and differentiability result

About lipschitz continuity of differentiable functions the following holds.

Theorem 1.

Let X,Y be Banach spacesMathworldPlanetmath and let A be a convex (see convex set), open subset of X. Let f:A¯Y be a function which is continuousMathworldPlanetmathPlanetmath in A¯ and differentiableMathworldPlanetmath in A. Then f is lipschitz continuous on A¯ if and only if the derivativePlanetmathPlanetmath Df is bounded on A i.e.


Suppose that f is lipschitz continuous:


Then given any xA and any vX, for all small h we have


Hence, passing to the limit h0 it must hold Df(x)L.

On the other hand suppose that Df is bounded on A:


Given any two points x,yA¯ and given any αY* consider the function G:[0,1]


For t(0,1) it holds


and hence


Applying Lagrange mean-value theorem to G we know that there exists ξ(0,1) such that


and since this is true for all αY* we get


which is the desired claim. ∎

Title Lipschitz condition and differentiability result
Canonical name LipschitzConditionAndDifferentiabilityResult
Date of creation 2013-03-22 13:32:42
Last modified on 2013-03-22 13:32:42
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 5
Author paolini (1187)
Entry type Result
Classification msc 26A16