TITLE: Function of two variables as a function of a function
QUESTION [5 upvotes]: Consider a function $f : \mathcal X \times \mathcal Y \mapsto \mathbb R$. I want to define $g_x(y) = f(x,y) : \mathcal Y \mapsto \mathbb R$. I want to say that 

$g_x$ is a ___ of function $f$.

What is the appropriate word for _____

REPLY [8 votes]: I've seen your $g_x$ called the $x$-section of $f$.  E.g. Folland's Real Analysis, section 2.5.
Edit: Another notation that's often useful is to write $f(x, \cdot)$ instead of $g_x$.

REPLY [6 votes]: This is sometimes called currying.  It is closely related to the notion of an exponential object.  But you don't really need to use either of these terms to perform this construction.
Edit:  Ah, I was assuming you were varying $x$.  If $x$ is fixed, you might want to call $g_x$ a restriction of $f$.