Vtyjkyuk

I have only seen the notation $f : A rightarrow B$ used for discussing functions. Can this notation be used for relations that are not functions?

For example, $tan : mathbb R rightarrow mathbb R$ appears to define a relation that is not left-total (for example $frac pi 2in mathbb R$ is not mapped to anything in codomain $mathbb R$) and hence not a function. Would it be incorrect to express this relation, that is not a function, using this notation?

Edit: To borrow a great example from Arthur's answer below, consider the notation $le; : mathbb R rightarrow mathbb R$. This seems to me to unambiguously express a relation that, while left-total is certainly not functional (e.g. $(3, 3), (3, 4) in ;le$) and hence not a function. If it's unambiguous, surely this notation can be used to express this relation that is not a function...

I guess I could express my question more clearly as: is there anything in the expression $f:Arightarrow B$ that asserts that $f$ is not merely a relation but specifically a function?

There is a category of relations having sets as objects and binary relations as arrows.

In that context the notation: $$R:Ato B$$ where $Rsubseteq Atimes B$, is fine.

In my view (somewhat coloured by categories) the expression $f:Ato B$ on its own is not enough to state that we are dealing with a function (or a relation).

I do recognize it as an arrow in a category and the context must make clear which category.

So my answer to your last question is: "no".

An entirely correct way would be $tansubseteq Bbb RtimesBbb R$. I think $tan: Bbb RtoBbb R$ would be considered technically incorrect but in many circumstances within allowable abuse of notation, mostly because $tan$ is still a partial function defined on most of the given domain. Writing something like $leq:Bbb RtoBbb R$ is probably taking it too far.