Differentiability a.e. and approximate differentiability a.e

Abstract

Let F be a finite real valued function defined on [0, 1]. We prove that F can be transformed into a function which is differentiable a.e. by a homeomorphic change of variables if and only if F is continuous on a dense set. We also show that F can be transformed into a function which is approximately differentiable a.e. if and only if each interval I ⊂ [ 0 , 1 ] I \subset [0,1] contains a nonempty perfect set P such that F | P F|P is continuous.