# Function of two variables as a function of a function

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 _____

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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 $g_x$.

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Thanks. I have been using your latter suggestion $f(x, \cdot)$ so far, but it gets unwieldy to use that every time. The term $x$-section sounds much better. I will check the exact definition in Folland's book. – Random User Oct 28 '10 at 17:08

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 $f$.

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Thanks. In what I am looking at $x$ does vary, so currying is the right notion. However, I am afraid of going to category theory (a fearsome beast for the intended audience) for something this elementary – Random User Oct 28 '10 at 17:06
@Random User: It's also a widely used operation in functional programming in computer science -- but I don't know if that would be familiar to your intended audience either. – Rahul Oct 28 '10 at 19:30