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I had this question in my mind for a long time but I was not sure if it makes sense to anyone.I would appreciate your valuable thoughts on this question.

  1. How to identify points of discontinuity of a function $f :\mathbb{R} \to \mathbb{R}$ given samples of the function (obtained by using Nyquist sampling or any other sampling technique with sampling frequency being as high as desired.)

  2. Is there any other alternate way of identifying points of discontinuity without evaluating the limit ?

EDIT 1: functions with a discontinuity are not strictly band-limited. But if we still go ahead by neglecting frequencies higher than certain limit, meaning bandlimiting, we observe the Gibb's phenomenon.

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    @Rahul Narain : the second part of the question is not really about sampled function but the actual function.2011-03-22

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No sampling method that leaves the samples greater than a certain distance apart can identify points of discontinuity unless you are given more information about the function. The simplest example is the step function:

$H(x) = \begin{cases} 0 \text{ if } x \le 0 \\ 1 \text{ if } x \gt 0 \end {cases}$

Unless you have samples approaching $x=0$ from above arbitrarily closely, you won't be able to tell this from a ramp

$s(x) = \begin{cases} 0 \text{ if } x \le 0 \\ 1/\delta \text { if } 0 \lt x \le \delta \\ 1 \text{ if } x \gt \delta \end {cases}$

as long as $\delta$ is less than your lowest positive sample point. If you know your function has no frequencies higher than a certain limit, you can rule it out.

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    You are right. The ramp I defined will also have arbitrarily high frequency components unless you round off the corners. So if you know your function is bandlimited, you know it is continuous-one result of the Nyquist theorem.2011-03-22