A note on $\varrho$-upper continuous functions

Abstract

In the present paper, we introduce the notion of classes of
$\varrho$-upper continuous functions. We show that $\varrho$-upper
continuous functions are Lebesgue measurable and, for
$\varrho<\frac{1}{2}$, may not belong to Baire class 1. We also prove
that a function with Denjoy property can   be non-measurable.