In some textbooks on analysis, I have encountered a definition of function/mapping that distinguishes the terminology mapping on $A$ to $B$ and mapping from $A$ to $B$; the first one refers to a function with domain $A$, the second to a function with domain $A'\subseteq A$. Furthermore, the notation $f: A\to B$ is sometimes used in the second meaning where the domain might be a proper subset of $A$.
My question is: is the second, nonstandard convention, just a local nuance of older books in Czech Republic where I live, or is/was there such convention in the past and/or in a larger geographical and cultural area?
As far as I'm concerned, the term function is used for the case where the entire $A$ is the domain. The concept you mention is usually called a 'partial function' from $A$ to $B$. When it comes to notation, I'm not quite sure if there's just one way to denote it. I think I've come across two different notations so far, neither of which I can now find in LaTeX symbol lists. One would be to write a vertical line through the $\to$ in $f:A\to B$ and the other would erase the middle part of the very same arrow to yield something like $f:A-\to B$, but not quite as long.