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I'm trying to think of a function which is measurable but not continuous a.e. My first thought was the Dirichlet function $\chi_{\Bbb Q}$ on $[0,1]$. That is to say

$$ \chi_{\Bbb Q}(x) = \begin{cases} 1 & \text{ if } x\in \Bbb Q \\ 0 & \text{ if } x\not\in\Bbb Q \end{cases}$$

and we restrict this function to $[0,1]$.

But then it seems to me that this function restricted to the irrationals is continuous because the restriction is constant. And $m([0,1]\smallsetminus ([0,1]\smallsetminus \Bbb Q))=0$ so the irrationals do seem to satisfy the the conditions which make $\chi_{\Bbb Q}$ continuous a.e.

Yet this post seems to indicate that $\chi_{\Bbb Q}$ cannot be continuous on any such restriction: https://math.stackexchange.com/a/67056/74378

I'm certainly misunderstanding something but I'm not sure what.

[Edit: Oh I think I see the discrepancy, which is that the person was asking for a function discontinuous a.e. So although $\chi_{\Bbb Q}$ is discontinuous a.e. because it's actually discontinuous everywhere, I guess it could also be continuous a.e. too?]

Addem
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    I think your confusion is stemming from the fact that when you restrict a function $f : X \to Y$ to a subspace $E$ of $X$ that $f|{E}$ being continuous does not mean that $f$ is continuous at every point in $E$. In this specific example, you're noticing that $\chi{\mathbb{Q}}|{\mathbb{R} - \mathbb{Q}}$ is continuous (as it is constant), but this does not necessarily mean that $\chi{\mathbb{Q}}$ (without the restriction) is continuous at any irrational numbers. So your example does indeed work. – jl00 Aug 16 '21 at 21:25
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    I think @jl00's comment addresses most of your concerns. For the last sentence in your edit: $\chi_{\Bbb{Q}}$ isn't continuous anywhere, so it is not continuous a.e. If you define $f : (0, 1) \to \Bbb{R}$ so that $f(x) = 0$ if $x$ is irrational and $f(p/q) = 1/q$, where $p, q$ are coprime, then $f$ is continuous on $(0, 1) \setminus \Bbb{Q}$ and discontinuous on $(0, 1) \cap \Bbb{Q}$, so it is continuous a.e., while it is not continuous on any open interval. – Rob Arthan Aug 16 '21 at 21:45

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