Locally Non-Measurable Functions with Many Continuity Points

It is well known that every continuous function on the space of real numbers is Lebesgue measurable. Also, if a function is continuous almost everywhere, then it is automatically measurable. Thus, local non-measurability seems, at first glance, to exclude continuity. But is this really the case? In this note we will show that actually every $G_\delta$-set with empty interior and locally big complement (in the sense of measure theory) can be prescribed as the set of continuity points of locally non-measurable functions.