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How to show that if

$g_{\epsilon}(x) = \sum_{m=1}^M = f(m/M)\chi_{[(m-1)/M,m/M)}(x)$

and

$x \in [(m-1)/M,m/M)$ then $g_{\epsilon}(x) = f(m/M)$ and not $g_{\epsilon}(x) = \sum_{m=1}^M f(m/M)$?

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    You got some indices wrong, some other typos, but there is nothing to show? This follows by the definition of the indicator function you cal kappa,2012-12-07
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    I do not see why $g_{epsilon}(x) = f(m/M)$ and not $g_{\epsilon}(x) = \sum_{n=1}^M f(m/M)$?2012-12-07
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    Kappa is the indicator function, right? If yes, it is 0 if not in its defining interval; so all other terms are 0. It is 1 on its defining interval, so you get the one constant on that interval.2012-12-07
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    Using x for the general functional relationship, and also to evaluate it at the point x might have thrown you off. Think of the second x as y in that interval.2012-12-07

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