For your example to work you need to restrict the domain of your functions to some compact e.g. 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?
This is a good post. I remember being so confused in a real analysis class when my professor started talking about how important it is that we restrict our attention to continuous linear functions (what on Earth was a discontinuous linear function supposed to be?). This post explains what's going on better than my professor or textbook did.
I agree with one of the other commenters that this part is not technically phrased accurately:
Because eg the derivative of f(x)=3√x at x=0 is +∞ despite the fact that it's continuous there.