In computer science and programming, one frequently wishes to create a process that takes a certain type of input and spits something (immaterial to this discussion) out. However, in many cases we also want to create an extremely similar process that takes in a different type of input (this difference is usually just a technicality in comparison to what the last process ate up) and spits out the same thing that the last process did.
We frequently name processes. In the case we're considering, we want to name the two processes above. Of course, different processes get different names; however, the two different process above don't really seem sufficiently different to warrant different names. It's just by the hair of a technicality that the two (or more) processes are different.
Some programming languages allow you to use the same name for these technically different processes. Some languages don't. However in languages that do, the practice of recycling the same name for slightly different purposes is called "overloading".
Is there a precise common language term for this idea of "overloading" that would carry the same idea without possibly implying other notions?
"Overload" is commonly interpreted to put too much weight on something instead of just adding more weight that the thing can still manage.
I can think of the terms "stretch" and "recycle," but the former loads too much metaphor and the latter carries an overtone that the previous purpose an object had is no longer going to be pursued. With an overloaded name, one can anticipate that the several different processes that it is used on will EACH be used forever and periodically.
I want to port this usage to mathematics. An example sentence that I want to alter is:
"We say that f is continuous at x (a point) if ... . To overload the term, we say that f is continuous on X (a set) if ... ."
Not all mathematics people are computer science people.