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Let $C$ be the Cantor set. I know that it is possible to construct a function that is continuous on $[0,1]\setminus C$ and discontinuous on $C$ by just considering the characteristic function on $C.$ But can we have it be the other way around?

Because $C$ is closed, it is a $G_\delta$ set, and so it could be the set of points at which a function is continuous. I'm having a hard time figuring out if we can construct such a function, because $U = [0,1] \setminus C$ is dense, so I'm guessing we'd want the function to "jump around wildly" on $U$ if that makes sense?

Is it true that such a function can be created, and if so, how would we do so?

quasicoherent_drunk
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    In fact, for every $G_\delta$ subset $G \subset \mathbb{R}$, there exists a function $f$ continuous on $G$ and discontinuous on $\mathbb{R} \setminus G$. See https://math.stackexchange.com/questions/67620/set-of-continuity-points-of-a-real-function?noredirect=1&lq=1 – Nate Eldredge Sep 25 '19 at 02:43

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Define $f(x)=d(x,C)$ if $x$ is rational, $f(x)=2d(x,C)$ if $x$ is irrational.

zhw.
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