SUMS OF QUASICONTINUOUS FUNCTIONS WITH CLOSED GRAPHS

Abstract

We prove that every real-valued $\mathcal{B}^*_1$ function $f$ defined on a~separable metric space $X$ is the sum of three quasicontinuous functions with closed graphs, and there is a $\mathcal{B}^*_1$ function which is not the sum of two quasicontinuous functions with closed graphs. Consequently, if $X$ is a separable metric space which is a Baire space in the strong sense, then the next three properties are equivalent: (1)$f$ is a $\mathcal{B}^*_1$ function, (2) $f$ is the sum of (at least) three quasicontinuous functions with closed graphs, and (3) $f$ is a piecewise continuous function.