Abstract. (1) For any $\alpha < \omega_1$ there exists a Borel measurable function $g\colon{\msbm R}\to{\msbm R}$ such that $g+f$ is a Darboux function (is almost continuous in the sense of Stallings) for every $f\in B_{\alpha }$. This solves a problem of J. Ceder. (2) There is a function $g$ that is universally measurable and has the Baire property in restricted sense such that $g+f$ is Darboux for every Borel measurable function $f$. (3) There is $g\colon{\msbm R}\to{\msbm R}$ such that $f+g$ is extendable for each $f\colon{\msbm R}\to{\msbm R}$ that is Lebesgue measurable (has the Baire property). (4) For every $\alpha < \omega_1$, each $f\in B_{\alpha }$ is the sum of two extendable functions $f_1,f_2\in B_{\alpha }$. This answers a question of A. Maliszewski.

AMS Subject Classification (1991): 04A15, 26A15, 28A05, 28A20

Keyword(s): Darboux functions, almost continuous functions, extendable functions, universal set, universal function, universal summand

Received October 16, 1997 and in revised form May 8, 1998. (Registered under 3300/2009.)