For your example to work you need to restrict the domain of your functions to some compact e.g. C1([0,1])because the uniform norm requires the functions to be bounded.
Hmm, is this really a substantive problem? Call it an “extended norm” instead of a norm and everything in the post works, right? My reasoning: An extended norm yields an extended metric space, which still generates a topology — it’s just that points which are infinitely far apart are in different connected components. Since you get a perfectly valid topology, it makes perfect sense for the post to talk about continuity. Or at least I think so; am I missing something?
Hmm, is this really a substantive problem? Call it an “extended norm” instead of a norm and everything in the post works, right? My reasoning: An extended norm yields an extended metric space, which still generates a topology — it’s just that points which are infinitely far apart are in different connected components. Since you get a perfectly valid topology, it makes perfect sense for the post to talk about continuity. Or at least I think so; am I missing something?
Perhaps! I’m not familiar with extended norms. But when you say “let’s put the uniform norm on C1(R)” warning bells start going off in my head 😅