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Suppose we have a continuous real-valued function $f$ on an interval $[a, b].$ Now suppose we choose a set of numbers $X$ from $[a, b]$ where we will create discontinuities. Let $g:X \rightarrow (-\infty, \infty) - \{f(x)\}.$ Now consider the function $h$ such that $h(x) = f(x)$ if $x \notin X$ and $h(x) = g(x)$ if $x \in X.$ How would we compute the integral of $h?$ I would assume that if $X$ were finite then the integral would be the same because it would be a change of an infinitesimal area. But what if $X$ were infinite? It seems to me that the answer might change if $X$ were an interval versus $X$ were not.

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    $\begingroup$ The buzzword here is "measure," see here: en.wikipedia.org/wiki/Lebesgue_measure . So long as $X$ has measure zero, the integral will remain the same $\endgroup$ – TomGrubb Jul 27 '17 at 3:49
  • $\begingroup$ Thanks @ThomasGrubb! $\endgroup$ – 伽罗瓦 Jul 27 '17 at 4:03
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To sum it up very briefly, a point has ZERO width. The area under something of zero width is 0 x f(x). Zero times anything is zero. It seems strange, but you can have a function with an infinite amount of removable discontinuities and still get an area under the curve. The area under the discontinuities still amounts to nothing.

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