locally linearize a CDF

Question

I have a sequence of discrete CDF's that converge to continuous CDF. Assume we call it $F_n(x)$. If say at some point, say $R$, $F_n$ is differentiable, then we can write $F_n(R+\xi) \approx F_n(R)+F_n'(R)\xi$. Now when $F_n(x)$ is discrete, the derivative is a sequence of delta function and I am not sure what this means. However, what I need for my purposes is that $F_n(R+\xi) \geq F_n(R)+M(R)\xi$, where $\xi > 0$. i.e. some sense of local bounded, so I came up with this kind of def. $M_R = \lim_{\gamma \to \infty}\sup_{0 < \xi < \gamma}\frac{F_n(R+\xi)-F_n(R)}{\xi}$. Of course, $F_n(x)$ is right-continuous. Does this make any sense, It seems like rather elementary need to approximate the derivative of a sequence of discrete RV that converge to continuous one.

Answer 1

The derivative of the CDF of a random variable $X$ can only exist at points $x$ such that $P[X=x]=0$, otherwise the CDF is not even continuous at $x$.