PARAMETRIC SEMICONTINUITY IMPLIES CONTINUITY

Abstract

Suppose $a\ne b$ are non-zero. We prove that if either $\lim {\inf }_{t\to 0-}(f(x+bt)-f(x+at))\ge 0$ or $\lim {\inf }_{t\to 0+}(f(x+bt)-f(x+at))\ge 0$ holds at each point of a set $E\subset \mathbf{R}$ which has the Baire property (is measurable), then $f$ is continuous at each point of $E$ apart from a first category set (a set of measure zero). We prove this via a generalization of the Uher-Thomson symmetric covering theorem.