On the Differences of Lower Semicontinuous Functions

Abstract

Answering one of the real function problems suggested by A. Maliszewski, the existence of a bounded Darboux function of the Sierpiński first class which cannot be expressed as a difference of two bounded lower semicontinuous functions is proved. As the reply to the other Maliszewski question, we show there exists an almost everywhere continuous Darboux function of the Sierpiński first class which is not a difference of two almost everywhere continuous lower semicontinuous functions.