Algorithmic Aspects of Lipschitz Functions
Cameron Freer, Bjø rn Kjos-Hanssen, André Nies, Frank Stephan · Computability · 2014
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 \(<![CDATA[$z$]]\).