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In this answer

https://math.stackexchange.com/a/1986000/344530,

the user arguments that there is only a finite number of points where $f_n(x) = 1$, thus integrable. I don't see how this actually works since I always thought that a function is continuous when it is continuous on every point. Now, the user shows that there are actually points where the function is not continuous, thus, it can't be continuous, can it?

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Julian
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    That answer never claims that the function is continuous everywhere. – Tobias Kildetoft Nov 01 '16 at 10:16
  • But wouldn't that be a necessary condition for the function being integrable? – Julian Nov 01 '16 at 10:17
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    No, it is not (and the answer even clearly states that it suffices to be continuous everywhere but at a finite number of points). – Tobias Kildetoft Nov 01 '16 at 10:18
  • But how can the function be integrable then? A continuous function is integrable, but a function that is continuous besides a finite number of points is integrable too? – Julian Nov 01 '16 at 10:21
  • Yes. I suggest you try to show this (it is a good exercise). I assume your definition of integrable is via Riemann sums? – Tobias Kildetoft Nov 01 '16 at 10:23
  • The first definition we got to know was the one where the upper sum and the lower sum converge against the same value, but yes, of course I know Riemann sums too, – Julian Nov 01 '16 at 10:25

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