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Can somebody please explain continuity and discontinuity in fourier series?

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This seems very broad, and it would be very difficult to guess where to start answering your question. Could you please elaborate, and indicate particular aspects of the theory you are having trouble with? – Jonas Meyer Nov 13 '10 at 19:38
If you mean a discontinuity like the square wave you can look at this picture – Américo Tavares Nov 13 '10 at 19:45
  1. If you have a removable discontinuity at a point, the Fourier series will converge to the limit of the function at the point.

  2. If you have a jump discontinuity at some point, then the Fourier series will converge to the average of the values of left and right limits. But the higher harmonics are significant, resulting in the "Gibbs phenomenon".

  3. For more complicated discontinuities, we really cannot say anything. You can have a look at the wikipedia page for more details.

The best one can say about the convergence of Fourier series is that it always converges almost everywhere, according to Carleson's theorem. But this kind of theorems involve more sophisticated levels of mathematical analysis.

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The first Wikipedia link is wrong; it points back to this page. – Hans Lundmark Nov 14 '10 at 6:48
It should perhaps be mentioned that some regularity assumptions are needed for these statements to be true. For example, in the case of a jump discontinuity, one commonly used sufficient condition for convergence is that the function have one-sided derivatives at the point in question. – Hans Lundmark Nov 14 '10 at 7:05
@Hans Lundmark: Thanks, I fixed the link. – user1119 Nov 14 '10 at 8:17

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