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.

  • 3
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.