Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. My professor defined a map $f: X \to Y$ and made clear that this was a map of "sets." However, he went on to talk about continuity of $f$ and the definition was in terms of these distance functions $d_X$ and $d_Y$. This doesn't make much sense to me, however, if these are just "maps" of sets. We could use any metric on the sets, realistically.
Do we define a function independently of the metric and then talk about continuity only with respect to some metric for $X$ and another metric for $Y$? Once we change either metric, continuity can change. So instead of saying "$f$ is continuous" we really want to say, "$f$ is continuous provided we use the metrics $d_X$ and $d_Y$." Is there a better shorthand convention?
Best Answer
Indeed, the metric can have an effect on whether the function is continuous or not. The same function can be continuous in one metric and not continuous in another. In fact, the only function you can really be sure is always continuous is the constant function.
But I do not think saying "$f$ is continuous" is wrong in any way, so long as it is clear from the surrounding text which metric we are using.
If you want to be more precise, you could say "$f$ is a continuous map from $(X, d_X)$ to $(Y, d_Y)$.
Note:
Sometimes, the shorthand
$$f:(X, d_X)\to(Y, d_Y)$$ is also used, and in terms of speaking about continuity, sure, it gets the point across, but I do not really like this notation, as it is overloading the existing map notation. $f$ is still a map from $X$ to $Y$, so it is correct to write $f:X\to Y$.