Function vs Partial Function

Question

Generally speaking, can we say that "any function is a partial function"? I know what is exactly a partial function, but is it correct to imply that arbitrary function can be seen as a partial function?

Answer 1

A partial function $f: Z \to Y$ is usually defined to be an ordinary function $f: X \to Y$ where $X \subseteq Z$. In your case, $Z=X$, so yes.

This would only go wrong if you defined a partial function $f: Z \to Y$ as an ordinary function $f: X \to Y$ for some subset $X \subset Z$, for then taking $X=Z$ would be prohibited. However, I have never seen that in the literature.