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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.

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    $\begingroup$ 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 $\endgroup$ – TheBridge Dec 20 '13 at 20:14
  • $\begingroup$ Why would that cause any trouble? And which other way would you prefer to use to define a function? $\endgroup$ – T. Eskin Dec 20 '13 at 20:14
  • $\begingroup$ @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. $\endgroup$ – Pratyush Sarkar Dec 20 '13 at 20:19
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    $\begingroup$ @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. $\endgroup$ – Andreas Blass Dec 20 '13 at 21:20
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    $\begingroup$ @PratyushSarkar They're the same function. (They are, however, different as morphisms in the category of metric spaces.) $\endgroup$ – Andreas Blass Dec 20 '13 at 22:12
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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.

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