Is it accurate to say that uniform continuity implies continuity but the converse may not hold, unless continuous over a closed interval in which case uniform continuity and continuity are equivalent?

I know the first part (the truth of only the forward implication and not the converse in the general case) is true from various sources, but I suspect the second part is true following Theorem 5.48 from Thomas/Bruckner Elementary Real Analysis:

Let f be continuous on [a, b]. Then f is uniformly continuous.

Obviously this contradicts the fact that the converse does not hold, so is this only true and the two definitions equivalent for a closed interval [a, b] and not any arbitrary set?