If $Y$ is a discrete random variable, and I define $F(x)=P(Y \leq x),$ where $x \in \mathbb{R},$ can I differentiate $F(x)$?
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2The short answer is no — so you might get a informative answer to a question that says more about what you want. – Matt F. Jan 12 '22 at 17:35
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1@Matt The long answer is yes (*e.g.*, $F$ often has a derivative almost everywhere and it has a generalized derivative everywhere; it often can be expressed as a Radon-Nikodym derivative with respect to a suitable (singular) measure; a "differential" $\mathrm{d}F$ is useful in Stieltjes integrals; and so on). But your recommendation is still an excellent one. – whuber Jan 12 '22 at 19:00
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@Quema ... What are you trying to do? – Glen_b Jan 12 '22 at 23:05
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I need to analyze if the rate of change of $F(x)$ is larger than $1/x$. – Quema Jan 13 '22 at 18:43
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In the (many) situations where $F$ is not differentiable, what exactly would you mean by the "rate of change of $F(x)$"? It sounds like you might be trying to analyze the tail properties of a distribution. If so, [our posts on heavy tailed distributions](https://stats.stackexchange.com/search?q=+long+heavy+tail+distribution) describe and illustrate effective techniques, many of which do not require differentiability. – whuber Feb 18 '22 at 21:02