Definition of functions on metric spaces.

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.

• I don't think anything happens at all, continuity is a property of a function it is not necessary for definition of a function, in the same way properties on space $X$ or $Y$ do not have any impact neither, unless I misunderstood you point. Best regards – TheBridge Dec 20 '13 at 20:14
• Why would that cause any trouble? And which other way would you prefer to use to define a function? – T. Eskin Dec 20 '13 at 20:14
• @ThomasE. I thought it would because you can define two functions $(f, X, Y)$ and $(f, X, Y)$ but with different metrics for the second one. Then the two functions are the same but the first may be continuous while the second is not. – Pratyush Sarkar Dec 20 '13 at 20:19
• @PratyushSarkar Indeed, one and the same function can be continuous between one pair of metric spaces and discontinuous between another pair. If you leave the underlying sets $X$ and $Y$ the same but change the metrics, then you are dealing with a different pair of metric spaces. – Andreas Blass Dec 20 '13 at 21:20
• @PratyushSarkar They're the same function. (They are, however, different as morphisms in the category of metric spaces.) – Andreas Blass Dec 20 '13 at 22:12

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 $Y$
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.