What are the assumptions that I need on the function $P( X | Y = y )$ (where $P$ is the conditional probability density function of X knowing an observation y of Y - so is bounded by one and always positive) in order to be able to write the following?
$$ \frac{\partial }{\partial \tau} \left( \int_{-\infty}^{\tau} P( X = x | Y = y ) \right) = P( X = \tau | Y = y ).$$
The partial derivative $$\frac{\partial}{\partial x}\int_{-\infty}^{x}f_{X|Y}(x^\prime|y) \mathrm{d}x^\prime=P(X=x|Y=y)=f_{X|Y}(X=x|Y=y)$$ always if the conditional distribution $F_{X|Y}(x|y)=\int_{-\infty}^{x}f_{X|Y}(x^\prime|y) \mathrm{d}x^\prime$ is differentiable at every $X=x, \forall y \in \Omega_y$; which simply means that the Conditional CDF has to be well defined to be differentiable everywhere. At those places where the derivatives do not exist, the distribution $f_{X|Y}$ cannot be defined as we want to and hence does not exist.