Heine–Cantor theorem

The set of Heine (after Eduard Heine, or set of Heine- Cantor ) from the real analysis makes a statement about continuous functions. It was proved in 1872 by Eduard Heine and named after him, according to Jürgen Heine this fact, however, was previously discovered by Karl Weierstrass.

Statement

If a function in the compact interval continuous, then it is there even uniformly continuous.

In other words, there is a so to any that for any two points in the interval, and comprising:

Evidence

A typical proof is by contradiction. Is not uniformly continuous, so there is a and every point, so that

By the theorem of Bolzano- Weierstrass has the bounded sequence has a convergent subsequence whose limit is included in the interval. This is because

Also limit of the sequence. The continuity of follows and. Therefore, there is a so that and for all. From this it follows

Contrary to all. Therefore, the assumption made was wrong and it follows the uniform continuity.

Generalization

With an almost identical proof to this theorem generalizes to compact metric spaces:

Is a compact metric space is a metric space and steadily, so is uniformly continuous.

Counterexample

For non- compact intervals of the set of Heine 's wrong. The function is continuous but not uniformly continuous. In fact there are not, that satisfies the condition of uniform continuity. Indeed, if any, there are with. Then follows

But

So can not be uniformly continuous.