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

thanks.

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The mystery symbol there is actually called \xi. –  Qiaochu Yuan Aug 26 '10 at 5:58
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We generally don't define $f(\xi+0)$ in this way: I've never seen this cheesy trick of defining one-sided limits using squares before. What is this book? –  Robin Chapman Aug 26 '10 at 6:18
    
@Chapman: It is from Introduction to Calculus and Analysis by R.Courant and F.John. –  Jichao Aug 26 '10 at 8:02
    
Concerning the title question, here is a quote from page 49 of Riemann's Zeta Function by H. Edwards: "In other words $\psi (x)=\sum_{p^{n}<x}\log p$ except when $x$ is a power; at the jumps $x=p^{n}$ the value of $\psi $\ is defined *as usual*, to be halfway between the new and old values $\psi (x)=\frac{1}{2}[\psi(x-\epsilon )+\psi (x+\epsilon )]$". Concerning your last question, I have never seen those definions in this way. Since $\epsilon ^2$ tends to zero fast than $\epsilon$ as $\epsilon$ approches zero, I guess that there is some connection with the derivative of function. –  Américo Tavares Aug 26 '10 at 9:42
    
@Robin Chapman:But if we do not use squares then $\epsilon$ could be negative. –  Jichao Aug 27 '10 at 12:52
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up vote 7 down vote accepted

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. –  AD. Aug 26 '10 at 6:00
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