# Who first proved the “Cantor-Heine theorem” on uniform continuity?

The theorem is that any continuous function on a compact is uniformly continuous. It is called "Heine", and sometimes also "Heine-Cantor" theorem.

My question is: what is the contribution of Cantor to this theorem, did Heine proved it after Cantor, or following Cantor's ideas? Was it proved before both Heine and Cantor?

"In 1872, Eduard Heine in «Die Elemente der Functionenlehre» gave a definition of the limit function using Cantor’s fundamental sequences. Every convergent sequence was represented as the sum of its limit and the elementary (decreasing) sequence. On this basis, Heine formulates the condition of continuity, the definition of uniform continuity in terms of $\epsilon-\delta$, the theorem of uniformly continuous functions and as a method of proof of there was cover lemma. The theorem of uniformly continuity was necessary for intervals between irrational number and its limit, the rational number. (It was formulated by Cantor, as Heine wrote. Now it is Cantor-Heine theorem)."