Does anyone know about how to prove the distribution function $F_{X}(x)$ for a continuous random variable X is differentiable almost everywhere?
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You can find a proof of the latter part here. – Hellen Sep 23 '17 at 02:14
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@Hellen I don't think it needs to have a finite number of discontinuities (consider the CDF of any distribution with support just on all the integers). Countable though. – spaceisdarkgreen Sep 23 '17 at 02:26
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@Hellen still even though the countable part would save the first part of your argument I think it might wreck (or at least complicate) the second. For instance it's possible for a nondecreasing bounded function to be discontinuous on a countable dense set of points like the rationals. This is somewhat moot since OP asked specifically about continuous RVs to which the theorem about non-decreasing continuous functions can be applied directly – spaceisdarkgreen Sep 23 '17 at 02:40