While I have come to understand that $f:a\mapsto b$ means that for input from the set $a$, the function will return a value from the set $b$, I am curious as to how far one may "drag" this notation. Say $f(x) = x$, would $f: 1 \mapsto 1$ be valid, or does one have to write $f: \mathbb{R} \mapsto \mathbb{R}$? (While I understand that my initial example would seldom prove useful, I am curious as to what is valid notation or not.)