almost continuous function

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