In the post Definition of functions, it is stated in the accepted answer that one way to define a function is to define it as the triple $(f, X, Y)$ where $f \subset X \times Y$. My question is what happens when we consider functions on spaces with more structure, e.g. metric spaces? If we define functions to be $(f, X, Y)$, then technically we are ignoring the metrics which comes along with the sets $X$ and $Y$ which is a problem when dealing with continuity etc.
Continuity is not a property of a function from set $X$ to set $Y$. It is a relation between a function, a metric on $X$, and a metric on $Y$. One can imagine a continuity checker as a machine into which you put
- a function from $X$ to $Y$
- a metric on $X$
- a metric on $Y$
and it signals "continuous" or "not continuous". Naturally, if you change some of the inputs, such as the metrics, the result may change.
In practice one often deals with sets on which the metric is understood without saying what it is, e.g., the real line $\mathbb R$. And so we say that $f:\mathbb R\to\mathbb R$ is continuous, because the other two ingredients that go into the continuity checker are so common that they need not be mentioned.