Gauges of Baire class one functions

Abstract

Let K be a compact metric space and f : K → R be a bounded Baire class one function. We proved that for any ε > 0 there exists an upper semicontinuous positive function δ of f with finite oscillation index and |f (x) − f (y)| < ε whenever d(x, y) < min{δ(x), δ(y)}.

The first homotopic Baire class of maps with values in ANR's coincides with the first Baire class

“…These results were generalized by many mathematicians in different ways (see [7], [6], [1] and the literature given there). The aim of the present paper is to introduce and study a homotopy counterpart of Baire-one maps.…”