TITLE: In written mathematics, is $f(x)$a function or a number?
QUESTION [23 upvotes]: I often see notation/wording like "let $f(x)$ be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$".
I would say that $\sin$ and $x \mapsto \sin(x)$ are functions, while $\sin(x)$ is a real number.
Are there any correct or best practices in this regard?

REPLY [4 votes]: Strictly speaking, I would say that $f(x)$ is the value of the function $f$ evaluated at $x$. However, $\frac{x^3-2}{x+1}$ might be used as a function; it is probably because $x\mapsto\frac{x^3-2}{x+1}$ is harder to write and takes up more space.