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Suppose $X$ is a random variable with distribution $F$. Let $\mathcal{X}_{d} = \{x : F(x) - F(x-) > 0\}$ denote the set of discontinuities of $F$, and let $\mathcal{X}_{c}$ be the complement of $\mathcal{X}_{d}$. Define $F_{d}(x) = P[X \leq x, X \in \mathcal{X}_{d}]$ and $F_{c}(x) = P[X \leq x, X \in \mathcal{X}_{c}]$, so that $F(x) = F_{d}(x) + F_{c}(x)$.

I would like to think of the probability that another random variable $Y$ is in some set $A$ conditional on the event that $X = x$. If $x \in \mathcal{X}_{d}$, this seems straightforward: $P[Y \in A \vert X = x] = P[Y \in A, X = x]/P[X = x]$ which is well defined since $P[X = x] > 0$ for $x \in \mathcal{X}_{d}$.

But what is the proper way to write $P[Y \in A \vert X = x]$ for $x \in \mathcal{X}_{c}$? Is it valid to say $P[Y \in A \vert X = x] = \frac{\partial}{\partial x}P[Y \in A, X \leq x]/\frac{\partial}{\partial x}F_{c}(x)$, assuming that $F_{c}(x)$ is differentiable at $x$? If not, what is the correct expression?

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