I'm self teaching myself probability theory and while reading through 'Introduction to Probability' by Blitzstein, I got a question regarding notation.
If Y is a discrete r.v. and X is a continuous r.v., then $P(Y = y|X = x)$ is used in the book and the author briefly talks about why $X = x$ is used here. He says
"Rigorously speaking, we are actually conditioning on the event that $X$ falls within a small interval of $x$, say $X \in (x-\epsilon,x+\epsilon)$ and then taking a limit as $\epsilon$ approaches $0$." - p.288
If so, my question is if I can do the same for continuous r.v.s. For example if $Z$ is a continuous r.v., could I say $f_{X,Z}(X=x,Z=z)$ instead of $f_{X,Z}(x,z)$ ?
No, don't write $f_X(x)=f(X=x)$ because the right-hand side does not make clear sense. Instead, for a continuous R.V. you can write $$F_X(x)=P(X\leqslant x)$$ where $F_X(x)$ is the CDF.