Heine–Cantor theorem

In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if "M" is a compact metric space, then every continuous function

:"f" : "M" → "N",

where "N" is a metric space, is uniformly continuous.

For instance, if "f" : ["a","b"] → R is a continuous function, then it is uniformly continuous.

Proof

Suppose that "f" is continuous on a compact metric space "M" but not uniformly continuous, then the negation of :$forall\; varepsilon\; >\; 0\; quad\; exists\; delta\; >\; 0$ such that for all "x", "y" in "M"

is:

:$exists\; varepsilon\_0\; >\; 0$ such that $forall\; delta\; >\; 0\; ,\; exists\; x,\; y\; in\; M$ such that and $ho\; (f(x)\; ,\; f(y)\; )\; ge\; varepsilon\_0$.where "d" and $ho$ are the distance functions on metric spaces "M" and "N", respectively.

Choose two sequences "x"_"n" and "y"_"n" such that: and $ho\; (\; f\; (x\_n),\; f\; (y\_n))\; ge\; varepsilon\_0$.

As the metric space is compact there exist two converging subsequences ($x\_\{n\_k\}$ to "x"₀ and $y\_\{n\_k\}$ to "y"₀), so:but as "f" is continuous and $x\_\{n\_k\}$ and $y\_\{n\_k\}$ converge to the same point, this statement is impossible.

For an alternative proof in the case of $M\; =\; [a,b]$ a closed interval, see the article on non-standard calculus.

ee also

* Georg Cantor

External links

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