On natural functions and Lipschitz functions.

Abstract

We give certain characterizations of the so-called natural functions introduced by Chistyakov. We show that the set of natural functions from $[a, b]$ into a normed space $X$ is small in the porosity sense in the set of uniform Lipschitz 1 functions with the norm inherited from the space of $BV$ functions. The sizes of other classes of Lipschitz functions in $BV$ are also established.