Recently I am studying real analysis, when I read about the introduction part of stein's 《real analysis》, I get really confused about the first example he gave in page xvi. From where we can see Riemann integration has some limitations and need to be refined? Is the problem lies just at the incompleteness of Riemann integrable functions space? If so ,why did he introduced fourier series and parseval's identity rather than give us an exact example to show the incompleteness of the space. I think I do not understand this problem clearly, please give me a more detailed explanation, thanks very much.
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any unbounded function is not Riemann integrable, also functions with too much discontinuities aren't either. Take a look here for a classic example. Also, try to use the search bar, by example see here also – Masacroso Mar 05 '24 at 09:48