On the Property N-1

Abstract

We construct a continuous function $f:[\mathrm{0,1}]\to \mathbb{R}$ such that $f$ possesses $N^{-\mathrm{1}}$-property, but $f$ does not have approximate derivative on a set of full Lebesgue measure. This shows that Banach’s Theorem concerning differentiability of continuous functions with Lusin’s property $(N)$ does not hold for $N^{-\mathrm{1}}$-property. Some relevant properties are presented.