Do we use "subdomain" terminology in order to specify that $Y$ is subset of domain $X$?

Question

I have very simple question.

For simplicity, let's take into consideration a function $f(x)$ defined over domain $X$. I consider some set $Y$, that is $Y\subset X$. Can I say that "$Y$ is subdomain of domain $X$"? Please advise.

Answer 1

No, this is not usually done. One would rather talk about the restriction of $f$ to $Y$. If $f:X \to Z$, this restriction is often denoted by $f\rvert_Y:Y \to Z$.