Heine–Cantor theorem

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.

Proof

Suppose that M and N are two metric spaces with metrics d_M and d_N, respectively. Suppose further that $f:M\to N$ is continuous, and that M is compact. We want to show that f is uniformly continuous, that is, for every $\epsilon >0$ there exists $\delta >0$ such that for all points x,y in the domain M, $d_{M}(x,y)<\delta$ implies that $d_{N}(f(x),f(y))<\epsilon$ .

Fix some positive $\epsilon >0$ . Then by continuity, for any point x in our domain M, there exists a positive real number $\delta _{x}>0$ such that $d_{N}(f(x),f(y))<\epsilon /2$ when y is within $\delta _{x}$ of x.

Let U_x be the open $\delta _{x}/2$ -neighborhood of x, i.e. the set

U_{x}=\left\{y\mid d_{M}(x,y)<{\frac {1}{2}}\delta _{x}\right\}

Since each point x is contained in its own U_x, we find that the collection $\{U_{x}\mid x\in M\}$ is an open cover of M. Since M is compact, this cover has a finite subcover. That subcover must be of the form

U_{x_{1}},U_{x_{2}},\ldots ,U_{x_{n}}

for some finite set of points $\{x_{1},x_{2},\ldots ,x_{n}\}\subset M$ . Each of these open sets has an associated radius $\delta _{x_{i}}/2$ . Let us now define $\delta =\min _{1\leq i\leq n}{\frac {1}{2}}\delta _{x_{i}}$ , i.e. the minimum radius of these open sets. Since we have a finite number of positive radii, this number $\delta$ is well-defined. We may now show that this $\delta$ works for the definition of uniform continuity.

Suppose that $d_{M}(x,y)<\delta$ 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 $d_{M}(x,x_{i})<{\frac {1}{2}}\delta _{x_{i}}$ . The Triangle Inequality then implies that

d_{M}(x_{i},y)\leq d_{M}(x_{i},x)+d_{M}(x,y)<{\frac {1}{2}}\delta _{x_{i}}+\delta \leq \delta _{x_{i}}

implying that x and y are both at most $\delta _{x_{i}}$ away from x_i. By definition of $\delta _{x_{i}}$ , this implies that $d_{N}(f(x_{i}),f(x))$ and $d_{N}(f(x_{i}),f(y))$ are both less than $\epsilon /2$ . Applying the Triangle Inequality then yields the desired