It's a standard result that the continuous locus is always G-delta. For each r>0, let U(r) be the set of points x such that some neighborhood of x maps into some ball of radius r. Then each U(r) is open, and the continuous locus is their intersection. Conversely, given a G-delta set, I'm pretty sure it's not hard to construct a function with that continuous locus, though I don't remember how off the top of my head.

It's a standard result that the continuous locus is always $G_\delta$. For each $r>0$, let $U(r)$ be the set of points $x$ such that some neighborhood of $x$ maps into some ball of radius $r$. Then each $U(r)$ is open, and the continuous locus is their intersection. Conversely, given a $G_\delta$ set, I'm pretty sure it's not hard to construct a function with that continuous locus, though I don't remember how off the top of my head.

It's a standard result that the continuous locus is always G-delta. For each r>0, let U(r) be the set of points x such that some neighborhood of x maps into some ball of radius r. Then each U(r) is open, and the continuous locus is their intersection. Conversely, given a G-delta set, I'm pretty sure it's not hard to construct a set with that continuous locus, though I don't remember how off the top of my head.

It's a standard result that the continuous locus is always G-delta. For each r>0, let U(r) be the set of points x such that some neighborhood of x maps into some ball of radius r. Then each U(r) is open, and the continuous locus is their intersection. Conversely, given a G-delta set, I'm pretty sure it's not hard to construct a function with that continuous locus, though I don't remember how off the top of my head.