Is there notation for an function being undefined?

Question

I could think of $\lnot\exists x. f(y)=x$, but that seems hacky.

Answer 1

In mathematical logic, especially computability theory, the up-arrow "$\uparrow$" (LaTeX code "\uparrow") is used this way when talking about partial functions; if $p$ is a partial function and is not defined on input $e$, we write "$p(e)\uparrow$."

This brings with it a notion of equality for partial functions/expressions, for which we use the symbol "$\simeq$" (LaTeX code "\simeq"). For individual expressions $x,y$ we say $x\simeq y$ iff either $x$ and $y$ are both undefined, or $x$ and $y$ are both defined and are equal, and for partial functions $p,q$ we say $p\simeq q$ if $p(i)\simeq q(i)$ for all valid inputs $i$.

However, I haven't seen this notation used this way outside of logic.