Let $A$ be a Borel set with positive but not full measure in every 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.

As suggested in a comment by Pietro Majer: Let $A$ be a Borel set with positive but not full measure in every 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.