In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if f : M → N is a continuous function between two metric spaces, and M is compact, then f is uniformly continuous. An important special case is that every continuous function from a closed interval to the real numbers is uniformly continuous.
Suppose that M and N are two metric spaces with metrics dM and dN, respectively. Suppose further that is continuous, and that M is compact. We want to show that f is uniformly continuous, that is, for every there exists such that for all points x,y in the domain M, implies that .
Fix some positive . Then by continuity, for any point x in our domain M, there exists a positive real number such that when y is within of x.
Let Ux be the open -neighborhood of x, i.e. the set
Since each point x is contained in its own Ux, we find that the collection is an open cover of M. Since M is compact, this cover has a finite subcover. That subcover must be of the form
for some finite set of points . Each of these open sets has an associated radius . Let us now define , i.e. the minimum radius of these open sets. Since we have a finite number of positive radii, this number is well-defined. We may now show that this works for the definition of uniform continuity.
Suppose that for any two x,y in M. Since the sets form an open (sub)cover of our space M, we know that x must lie within one of them, say . Then we have that . The Triangle Inequality then implies that
implying that x and y are both at most away from xi. By definition of , this implies that and are both less than . Applying the Triangle Inequality then yields the desired
For an alternative proof in the case of M = [a, b] a closed interval, see the article on non-standard calculus.
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