For obvious reasons coming from probability theory I have been calling the function $f_{x_0}(y)$ ($x_0$ fixed) "marginal function". However, reviewing some literature I've noticed that the word "marginal" relates mainly to areas of probability and economy, and not analysis, which is the area I am working on.
Is there any better way to call the function $f_{x_0}$, or is the term "marginal" correct? If so, could you please provide some reference where $f_{x_0}$ is referred to as a marginal function? (and not marginal distribution, which is a concept of probability theory).
I don't know whether there is a single word that describes exactly what you want, like "marginal", but in general terms I would say that $f_{x_0}(y)$ is the restriction of $f(x,y)$ to the line $x=x_0$. One common notation for this is $$ f(x,y)\big|_{x=x_0} $$ Similarly, for any subset $S\subseteq \Bbb R^2$, $f(x,y)$ induces a restricted function $f(x,y)|_S$. There is also, of course, nothing special about $\Bbb R^2$ in this case; the domain of $f$ might be any set, and we would still call this construction a restriction.