I'm writing a paper (in physics), and I want use the same name for two related functions that have different domains. Please allow me to elaborate.
I have function $f: R\longmapsto R$. I want to define a second function on the set of 2-tuples $f:R\times R \longmapsto R$ where f(x,x) = f(x)f(x). Here is my question: Can I use the same name for the two functions? In the context of this paper, it would reduce the amount of notation if I could give these two functions the same name. I would argue that there is no confusion here, because the two functions have different domains. So from the usage of the function, and the domain, a reader will always know which function being is referring to. Please comment. thanks.
It's not uncommon to use the same symbol when you restrict or enlarge the domain of a function, but that's not what you're doing here. Your function of two variables is not necessarily ever equal to the function of one variable: $f(x)$ might always be negative and $f(x,y)$ always positive. How would it look, notation-wise, if you used $F(x,y)$ for the two-variable function?