Heine's 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 proven by Eduard Heine in 1872 and named after him, but according to Jürgen Heine , this fact was discovered earlier by Karl Weierstrass .

statement

Heine's theorem says:

If a function is continuous in the compact interval , then it is even uniformly continuous there . ${\displaystyle f}$ ${\displaystyle [a,b]}$

In other words: for an arbitrary one exists such that for any two arbitrary places and from the interval with : ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta =\delta (\varepsilon )>0}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle [a,b]}$ ${\displaystyle |x_{2}-x_{1}|<\delta }$

${\displaystyle |f(x_{2})-f(x_{1})|<\varepsilon .}$

proof

Typical evidence is contradiction. If not uniformly continuous , there is one and for every point such that ${\displaystyle f}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x_{n},x_{n}'\in [a,b]}$

${\displaystyle \left|x_{n}-x_{n}'\right|<{\frac {1}{n}}}$ and ${\displaystyle \left|f(x_{n})-f(x_{n}')\right|\geq \varepsilon .}$

According to the Bolzano-Weierstrass theorem , the bounded sequence has a convergent subsequence whose limit is contained in the interval . This is because of ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (x_{n_{k}})_{k\in \mathbb {N} }}$ ${\displaystyle x}$ ${\displaystyle [a,b]}$

${\displaystyle \left|x_{n_{k}}-x_{n_{k}}'\right|<{\frac {1}{n_{k}}}}$

also limit value of the sequence . It follows from the continuity of and . Hence there is one such that and for everyone . It now follows ${\displaystyle (x_{n_{k}}')_{k\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle f(x_{n_{k}})\to f(x)}$ ${\displaystyle f(x_{n_{k}}')\to f(x)}$ ${\displaystyle k_{0}}$ ${\displaystyle \left|f(x_{n_{k}})-f(x)\right|<\varepsilon /2}$ ${\displaystyle \left|f(x_{n_{k}}')-f(x)\right|<\varepsilon /2}$ ${\displaystyle k\geq k_{0}}$

${\displaystyle {\begin{aligned}\left|f(x_{n_{k}})-f(x_{n_{k}}')\right|&=\left|(f(x_{n_{k}})-f(x))+(f(x)-f(x_{n_{k}}'))\right|\\&\leq \left|(f(x_{n_{k}})-f(x))\right|+\left|(f(x)-f(x_{n_{k}}'))\right|<\varepsilon /2+\varepsilon /2=\varepsilon \end{aligned}}}$

for all , contrary to for all . Hence the assumption made was wrong and the steady continuity follows. ${\displaystyle k\geq k_{0}}$ ${\displaystyle \left|f(x_{n_{k}})-f(x_{n_{k}}')\right|\geq \varepsilon }$ ${\displaystyle k}$

Generalization to compact metric spaces

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

If a compact metric space is a metric space and is continuous, then is uniformly continuous. ${\displaystyle K}$ ${\displaystyle M}$ ${\displaystyle f:K\rightarrow M}$ ${\displaystyle f}$

Another proof sketch for metric spaces

According to Otto Forster , for example, the theorem can also be proven on the basis of the Heine-Borel property - and without contradiction proof !

This proof can be sketched as follows:

Any one can be fixed for the compact metric space (with the metric ), the metric space (with the metric ) and the continuous mapping . For this purpose, what is required for the proof of uniform continuity must be determined. ${\displaystyle K}$ ${\displaystyle d_{K}}$ ${\displaystyle M}$ ${\displaystyle d_{M}}$ ${\displaystyle f}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \delta (\epsilon )}$

This is obtained by first the continuity property of attracts and from it to each one sets such that, for having always is met. ${\displaystyle f}$ ${\displaystyle a\in K}$ ${\displaystyle \delta (a)>0}$ ${\displaystyle b\in K}$ ${\displaystyle d_{K}(a,b)<\delta (a)}$ ${\displaystyle d_{M}(f(a),f(b))<{\frac {\epsilon }{2}}}$

Then you look at the out of sheer point environments existing open coverage to . Because of the compactness of , the Heine-Borel property means that a finite number of these environments already cover, for example for a certain one . ${\displaystyle K}$ ${\displaystyle (U_{\frac {\delta (a)}{2}})_{a\in K}}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle U_{\frac {\delta (a_{1})}{2}},\ldots ,U_{\frac {\delta (a_{n})}{2}}}$ ${\displaystyle n\in \mathbb {N} }$

Finally, one sets:

${\displaystyle \delta (\epsilon )=\min({\frac {\delta (a_{1})}{2}},\ldots ,{\frac {\delta (a_{n})}{2}})}$ .

The proof of the inequality occurring in the definition of uniform continuity is carried out using the triangle inequality .

Generalization to compact Hausdorff spaces

Heine's theorem can even be extended to any compact Hausdorff spaces beyond the compact metric spaces . This is a direct consequence of the fact that the topological structure of a compact Hausdorff space is subject to a clearly defined uniform structure . Their neighborhood system consists of all the surroundings of the diagonals in the associated product space , with those in open neighborhoods forming a fundamental system, which even gives a completely uniform structure . ${\displaystyle X}$ ${\displaystyle {\Phi }_{X}}$ ${\displaystyle \Delta =\{(x,x):x\in X\}}$ ${\displaystyle X\times X}$ ${\displaystyle X\times X}$

The following applies:

A steady mapping of the compact Hausdorff space into the uniform space is always uniformly continuous . ${\displaystyle f:X\rightarrow Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

Inference

A continuation sentence can be derived from Heine's theorem :

If it is a dense subset of the compact Hausdorff space and is a mapping of into the separated and complete uniform space , then is continuous and can be continued to a continuous mapping on exactly when - with regard to the uniform structure induced by on - is uniformly continuous. ${\displaystyle A\subseteq X}$ ${\displaystyle X}$ ${\displaystyle f:A\rightarrow Y}$ ${\displaystyle A}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle A}$

Counterexample

Heine's theorem is wrong for non-compact intervals. The function , is continuous, but not uniformly continuous. Indeed, there is none that satisfies the condition of uniform continuity. For if desired, so there are with . Then follows ${\displaystyle f:(0,1]\rightarrow \mathbb {R} }$ ${\displaystyle x\mapsto {\tfrac {1}{x}}}$ ${\displaystyle \varepsilon =1}$ ${\displaystyle \delta >0}$ ${\displaystyle \delta >0}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle {\tfrac {1}{n}}<\delta }$

${\displaystyle \left|{\frac {1}{n+1}}-{\frac {1}{n}}\right|={\frac {1}{n(n+1)}}<\delta }$,

but

${\displaystyle \left|f\left({\frac {1}{n+1}}\right)-f\left({\frac {1}{n}}\right)\right|=|(n+1)-n|=1\geq \varepsilon }$.

So it can not be uniformly continuous. ${\displaystyle f}$

literature

• Nicolas Bourbaki : General Topology (=  Elements of Mathematics . Part I). Addison-Wesley Publishing (et al.), Reading MA (et al.) 1966 ( MR0205210 ).
• Otto Forster : Analysis 2 . Differential calculus in R ⁿ , ordinary differential equations (=  Vieweg Studium ). 6th, revised and expanded edition. Vieweg Verlag, Wiesbaden 2005, ISBN 3-528-47231-6 .
• Jürgen Heine : Topology and Functional Analysis . Basics of abstract analysis with applications. 2nd, improved edition. Oldenbourg Verlag, Munich 2011, ISBN 978-3-486-70530-0 .
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
• Stephen Willard: General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA (et al.) 1970 ( MR0264581 ).

Individual evidence

1. ↑ Eduard Heine: The elements of the theory of functions. In: Journal for pure and applied mathematics . Vol. 74, 1872, pp. 172-188 .
2. ↑ Jürgen Heine: Topology and Functional Analysis. Oldenbourg, Munich 2002, ISBN 3-486-24914-2 .
3. ↑ Otto Forster: Analysis 2. 2005, p. 34.
4. ↑ Horst Schubert: Topology. 1975, p. 128 ff.
5. ↑ Nicolas Bourbaki: General Topology. Part I. 1966, p. 198 ff.
6. ↑ ^{a b} Horst Schubert: Topology. 1975, p. 129.
7. ^ ^{A b} Nicolas Bourbaki: General Topology. Part I. 1966, p. 201.
8. ↑ The neighborhood system of this induced uniform structure originates from the inclusion mapping and consists of the intersections of with the neighborhoods (Horst Schubert: Topologie. 1975, p. 110). ${\displaystyle A}$ ${\displaystyle i_{A\times A}\colon {A\times A}\rightarrow {X\times X}}$ ${\displaystyle A\times A}$ ${\displaystyle {\Phi }_{X}}$