Heine–Cantor theorem

{{Short description|Mathematical theorem}}

{{distinguish|Cantor's theorem}}{{No footnotes|date=April 2019}}

In mathematics, the Heine–Cantor theorem states that a continuous function between two metric spaces is uniformly continuous if its domain is compact.

The theorem is named after Eduard Heine and Georg Cantor.

{{Math theorem

|name=Heine–Cantor theorem

|If $f\; \backslash colon\; M\; \backslash to\; N$ is a continuous function between two metric spaces $M$ and $N$, and $M$ is compact, then $f$ is uniformly continuous.

}}

An important special case of the Cantor theorem is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous.

{{Math proof

|title=Proof of Heine–Cantor theorem

|Suppose that $M$ and $N$ are two metric spaces with metrics $d\_M$ and $d\_N$, respectively. Suppose further that a function $f:\; M\; \backslash to\; N$ is continuous and $M$ is compact. We want to show that $f$ is uniformly continuous, that is, for every positive real number $\backslash varepsilon\; >\; 0$ there exists a positive real number $\backslash delta\; >\; 0$ such that for all points $x,\; y$ in the function domain $M$, implies that .

Consider some positive real number $\backslash varepsilon\; >\; 0$. By continuity, for any point $x$ in the domain $M$, there exists some positive real number $\backslash delta\_x\; >\; 0$ such that when , i.e., a fact that $y$ is within $\backslash delta\_x$ of $x$ implies that $f(y)$ is within $\backslash varepsilon\; /\; 2$ of $f(x)$.

Let $U\_x$ be the open $\backslash delta\_x/2$-neighborhood of $x$, i.e. the set

:

Since each point $x$ is contained in its own $U\_x$, we find that the collection $\backslash \{U\_x\; \backslash mid\; x\; \backslash in\; M\backslash \}$ is an open cover of $M$. Since $M$ is compact, this cover has a finite subcover $\backslash \{U\_\{x\_1\},\; U\_\{x\_2\},\; \backslash ldots,\; U\_\{x\_n\}\backslash \}$ where $x\_1,\; x\_2,\; \backslash ldots,\; x\_n\; \backslash in\; M$. Each of these open sets has an associated radius $\backslash delta\_\{x\_i\}/2$. Let us now define $\backslash delta\; =\; \backslash min\_\{1\; \backslash leq\; i\; \backslash leq\; n\}\; \backslash delta\_\{x\_i\}/2$, i.e. the minimum radius of these open sets. Since we have a finite number of positive radii, this minimum $\backslash delta$ is well-defined and positive. We now show that this $\backslash delta$ works for the definition of uniform continuity.

Suppose that for any two $x,\; y$ in $M$. Since the sets $U\_\{x\_i\}$ form an open (sub)cover of our space $M$, we know that $x$ must lie within one of them, say $U\_\{x\_i\}$. Then we have that . The triangle inequality then implies that

:

implying that $x$ and $y$ are both at most $\backslash delta\_\{x\_i\}$ away from $x\_i$. By definition of $\backslash delta\_\{x\_i\}$, this implies that $d\_N(f(x\_i),f(x))$ and $d\_N(f(x\_i),\; f(y))$ are both less than $\backslash varepsilon/2$. 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 Non-standard calculus.