Can a Function Be Discontinuous Only at Irrationals?

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Here is this week's POTW:

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Does there exist a real-valued function on $\Bbb R$ that is discontinuous only on the irrationals?

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No one answered this week's problem. You can read my solution below.

Spoiler

No. If so an function $f$ existed, then its oscillation $\omega_f$ would be identically zero on $\Bbb Q$. The rationals can then be written as a countable intersection of open sets $A_n := \{x : \omega_f(x) < 1/n\}$. This implies $\Bbb Q$ is a G$_{\delta}$ set, in $\Bbb R$, contradicting the Baire category theorem.