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.

A subset $B$ of a Polish space $X$ is called Bernstein set if both $B$ and $X\backslash B$ meet every uncountable, closed subset of $X$. Such sets neither are Lebesgue measurable nor have the property of Baire, and as such, are of interest to analysts, topologists and descriptive set theorists alike. Taking advantage of the algebraic structure of ${ℝ}^k$, we exhibit a family of $2^{\mathrm{C}}$ homogeneous Bernstein sets in $ℝ$. We also show that if $X$ and $Y$ are uncountable Polish spaces and $f:X\to Y$ is continuous, then Bernstein sets are preserved under inverse images if and only if $f$ is at most $\omega$-to-$1$.