ON UNIVERSALLY BAD DARBOUX FUNCTIONS

Abstract

It is well known that the sum (and the product) of a continuous function and a Darboux function need not be Darboux in general ([9]). More precisely, for every nowhere constant continuous $g:R\to R$ there exists some “bad” Darboux function $f:R\to R$ such that $f+g$ or $f\cdot g$ do not have the Darboux property, see [2], [8]. It is the purpose of the present paper to construct a “universally bad” Darboux function $f$, see Corollary 2 below.