# almost continuous function

Let $m$ denote Lebesgue measure, $A$ be a Lebesgue measurable subset of $\mathbb{R}$, and $f:A\to\mathbb{C}$ (or $f:A\to\mathbb{R}$). Then $f$ is almost continuous if, for every $\varepsilon>0$, there exists a closed subset $F$ of $\mathbb{R}$ such that $F\subseteq A$, $m(A-F)<\varepsilon$, and $f|_{F}$ is continuous.

Title almost continuous function AlmostContinuousFunction 2013-03-22 16:13:45 2013-03-22 16:13:45 Wkbj79 (1863) Wkbj79 (1863) 4 Wkbj79 (1863) Definition msc 28A20 almost continuous LusinsTheorem2