When defining a function, mathematicians often write something like this:
Let $f\colon \mathbb R\to \mathbb R$ be the function given by $x\mapsto x^2$.
The purpose of this definition may be to assert something about this function, for example:
Then $f$ neither injective nor surjective.
One can shorten this whole thing as follows:
The function $f\colon \mathbb R\to\mathbb R:x\to x^2$ is neither injective nor surjective.
My question is whether $f\colon \mathbb R\to\mathbb R:x\to x^2$ is a correct notation. Also, I wonder if there are alternative notations. Maybe using the "$;$" instead of "$:$"?