On the maximum of two unilaterally continuous regulated functions.

Abstract

We prove that if $f$ is the maximum of two unilaterally continuous regulated functions, then the set $D_{un}(f)=\{x:f$ is not unilaterally continuous at $x\}$ is unilaterally isolated and for $x \in D_{un}(f)$ the inequality $f(x)<\max(f(x+),f(x-))$ holds. Moreover, for a regulated function $f$ such that $D_{un}(f)$ is isolated and for $x \in D_{un}(f)$ the inequality $f(x)<\max(f(x+),f(x-))$ holds, there are two unilaterally continuous regulated functions $g$, $h$ with $f=\max(g,h)$.