Yesterday, Randall Holmes mentioned to me the following nice characterization of continuity for functions between Euclidean spaces.
Theorem. A function is continuous iff it preserves path-connectedness and compactness.
This is an easy exercise, but I don’t remember having seen the characterization before, so I figured I could as well write down the argument I found. It is clear that the result holds for a much wider class of spaces than the , but to keep this post simple, I’ll just leave it this way.
Proof. For Euclidean spaces, continuity and sequential continuity coincide. Towards a contradiction, assume converges to but does not converge to .
- Case 1. The range of is infinite.
This quickly leads to a contradiction since is a compact set: We may as well assume that the map is injective, and since does not converge to we may in fact assume that all the stay away from . The set has an accumulation point, which cannot be so it must be for some . But then the set is both compact and lacks one of its accumulation points, contradiction.
- Case 2. The range is finite.
We may as well assume all have the same image. Fix paths in that we can combine to get a path (for example, could simply be a segment). By preservation of path connectedness, any with range of size at least 2 in fact has range of size continuum. If infinitely many of the have infinite range, one easily reduces to Case 1. So we may assume all the have constant range, but then has a disconnected image.