In $\mathbb{R}^n$, does continuity on paths imply continuity?

Question

Let $U \subset \mathbb{R}^n$ be open and connected (equiv. path connected), and let $f : U \to \mathbb{R}$. Is it true that if $f$ is continuous on all paths, that is, if for all continuous $\gamma : [0,1] \to U$, $f \circ \gamma$ is continuous, then $f$ must be continuous?

If this is true, could we weaken the hypotheses on $U$ in any nice way?

What did you try to solve this? – Moishe Kohan Feb 13 '22 at 19:59 — Moishe Kohan, Feb 13 '22 at 19:59

score 1 · Answer 1 · answered Feb 13 '22 at 19:59

1

Yes, this is true, and is a special case of the result proved in this answer: https://math.stackexchange.com/a/2446199/19006

answered Feb 13 '22 at 19:59

diracdeltafunk

16,589

So it's a dupe? – Randall Feb 13 '22 at 20:46
Instead of answering like that one should vote to close as a duplicate. – Moishe Kohan Feb 13 '22 at 23:50
Noted, thanks :) – diracdeltafunk Feb 14 '22 at 00:42

In $\mathbb{R}^n$, does continuity on paths imply continuity?

1 Answers1