This is in the context of the cumulative distribution function (CDF) of a random variable and its corresponding probability density function (PDF).
Here, the important thing to note is that the CDF has a countable number of jumps and/or discontinuities.
For example, for a random variable $X$, it is given that
$F_{X}(x) = \begin{cases}0&,& x <-1\\ \dfrac{x+2}{4}&,& -1\leq x <1\\ 1&,& x\geq 1\end{cases}$
Here, we see that that $F_{X}$ has two jumps: at $x = -1$ and also at $x = 1$, since $F_{X}((-1)^-) = 0$ and $F_{X}(-1) = \frac{1}{4}$, and also $F_{X}(1^-) = \frac{3}{4}$ and $F_{X}(1) = 1$.
In situations like these (i.e., where the CDF has a countable number of jumps and/or discontinuities), does the PDF exist?
Also, if the PDF does exist, then how can we find the PDF of the random variable?