Let $f$ be a bounded function on a close interval, $[0,1]$ e.g.. Can it be everywhere discontinuous and integrable?
Thanks!
P.S. It isn't a homework for me and I asked this question just out of curiosity.
Let $f$ be a bounded function on a close interval, $[0,1]$ e.g.. Can it be everywhere discontinuous and integrable? Thanks! P.S. It isn't a homework for me and I asked this question just out of curiosity. 


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Dpends on which type of integration you are asking for. If you mean Lebesgue integral, then yes, a function can be nowhere continuous but still have a Lebesgue integral, as Henr.L points out. If you want Riemann integral, then the answer is no. A theorem sometimes called Lebesgue's theorem states that a bounded function on the reals has a Riemannintegral if and only if the set of points where it is discontinuous has zero measure. The proof of this theorem is quite long but not technically difficult (in my opinion). 


Take the characteristic function of $\mathbb Q$: it is not Riemann integrable, everywhere discontinuous, almost everywhere 0 and thus Lebesgue integrable with integral 0. 


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