Algorithmic Aspects of Lipschitz Functions

Abstract

We characterize the variation functions of computable Lipschitz functions. We show that a real $<![CDATA[$z$]]$ is computably random if and only if every computable Lipschitz function is differentiable at $<![CDATA[$z$]]$ . Beyond these principal results, we show that a real $<![CDATA[$z$]]$ is Schnorr random if and only if every Lipschitz function with $<![CDATA[$L_{1}$]]$ -computable derivative is differentiable at $<![CDATA[$z$]]$ .