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.