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If $X$ is a Baire space and $f : X \to R$, then the set $D(f)$ of points where $f$ is discontinuous is an $F_{\sigma}$ set. Prove that $f$ is continuous on a dense subset of $X$ if and only if $D(f)$ is of first category in $X$.

I have proved that the set $D(f)$ of points where $f$ is discontinuous is an $F_{\sigma}$ set.

Need help in the second part, $f$ is continuous on a dense subset of $X$ if and only if $D(f)$ is of first category in $X$.

Can anyone give some hints in this problem.

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For the forward direction, since you know $D(f)$ is $F_\sigma$, you can write $D(f) = \bigcup_{n=1}^\infty E_n$ where each $E_n$ is closed. Let $C(f) = D(f)^c$ be the set of points at which $f$ is continuous. Note $C(f)$ is disjoint from each $E_n$. So if $C(f)$ is dense, show as a consequence that each $E_n$ is nowhere dense.

For the backward direction, if $D(f)$ is first category (meager), then $C(f)$ is comeager, and the Baire category theorem states that every comeager set is dense.

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