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## Main Question or Discussion Point

I am trying to prove a theorem related to the convergence of Fourier series. I will post my proof below, so first check it and then my question will make sense.

Is there any flaw in my proof? Also, here I proved it for integrable functions monotonic on an interval on the left of 0. But what if the function was not monotonic on any interval around 0, and was not Lipschitz continuous either? For example, f(x) = √|x|*sin(1/x). Can I still use step functions to approximate it as in the proof for monotonic functions? I know it oscillates wildly, but by taking sufficiently small intervals, it would seem that the limit would still go to f(0-) over 2. And yet I know this is not possible since there are continuous functions whose Fourier series do not converge at some points of continuity, and my proof would lead to a contradiction if it were true for all non-monotonic functions. So is my proof flawed, and if not, what prevents it from working for functions like √|x|*sin(1/x)?

Is there any flaw in my proof? Also, here I proved it for integrable functions monotonic on an interval on the left of 0. But what if the function was not monotonic on any interval around 0, and was not Lipschitz continuous either? For example, f(x) = √|x|*sin(1/x). Can I still use step functions to approximate it as in the proof for monotonic functions? I know it oscillates wildly, but by taking sufficiently small intervals, it would seem that the limit would still go to f(0-) over 2. And yet I know this is not possible since there are continuous functions whose Fourier series do not converge at some points of continuity, and my proof would lead to a contradiction if it were true for all non-monotonic functions. So is my proof flawed, and if not, what prevents it from working for functions like √|x|*sin(1/x)?