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My question is why do we define $f(\xi+0)$ as $\lim\limits_{\varepsilon \rightarrow 0} f(\xi + \varepsilon^2)$ and $f(\xi-0)$ as $\lim\limits_{\varepsilon \rightarrow 0} f(\xi - \varepsilon^2)$.

Thanks.

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    @Robin Chapman:But if we do not use squares then $\epsilon$ could be negative.2010-08-27

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It is not that we assign that value: Fourier series simply converge to that independently of our desires!

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    Yes, or even better if $f$ is not given as such then we can redefine $f$ to have the mean value at the point of discontinuity and then $S[f]$ (the Fourier series of $f$) will converge to $f$ at that point.2010-08-26