Let $A$ be [a Borel set with positive but not full measure in every interval](https://math.stackexchange.com/questions/57317/construction-of-a-borel-set-with-positive-but-not-full-measure-in-each-interval) in $(0,1)$ and let $f = 1_A$ be its indicator function.  Then if $g = f$ a.e., we have that $g$ takes both values 0 and 1 in every interval, and thus is nowhere continuous.