In mathematics and related technical fields, the term map or mapping is often a synonym for function. Along these lines, a partial map is a partial function, and a total map is a total function.
In many specific branches of mathematics, the term is used for a function with a specific property relevant to that branch, such as a continuous function in topology, a linear transformation in linear algebra, etc. (In Wikipedia, we always include a relevant adjective like "continuous" or "smooth" to avoid confusion.)
In formal logic, the term is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory.
See also: