Nearly, but not almost, continuous

Solution 1:

The characteristic function of a fat Cantor set gives an example (or any closed set with positive measure and empty interior). If $C$ is such a set, and $A$ is any null set, then $\mathbb{R}\setminus A$ contains an element $x$ of $C$. Every neighborhood of $x$ intersects $\mathbb{R}\setminus C$ in a set of positive measure (because nonempty open sets have positive measure), so for all $\delta>0$, $(x-\delta,x+\delta)\setminus A$ contains elements of $\mathbb R\setminus C$. This implies that the restriction of $\chi_C$ to $\mathbb{R}\setminus A$ is discontinuous at $x$.