This is question 4.6 from Rudin's book. The first problem I ran into when trying to show this is that I'm not really sure what the distance function is between $(x_1, f(x_1))$ and $(x_2,f(x_2))$. Doesn't this have to be defined?
2026-04-04 17:30:11.1775323811
Showing that a function on a compact metric space X is continuous iff its graph is compact
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In that chapter in the book, all continuity is between metric spaces. He does not mention how to define compactness for the graph in general but he does give the example that if we have a real-valued function on a domain $E \subseteq \mathbb{R}$, the graph can be seen as a subset of the plane (at least it says so in my edition), and as such has a natural topology. This leads me to think he does not want to see it as a general fact on metric spaces (this would expect the student to develop some theory and come up with a natural topology for the graph) but as sketched in this explanation: a real valued function on some subset of the reals.
Try to come up with a proof for that context. (Search on this site, it has been answered before, both by me and many others).