Theorem of minimum and maximum

A continuous function defined on [a, b] that takes its maximum and minimum

The principle of minimum and maximum is a mathematical theorem from the field of analysis , which is attributed to the German mathematician Karl Weierstrass . The theorem says that every real- valued and continuous function defined on a compact real interval is bounded and assumes its maximum and minimum in the domain of definition . It is one of the main theorems of analysis and is an important tool for proving the existence of extreme values ​​of such functions.

Theorem of minimum and maximum

The sentence can be formulated in several versions:

(Ia) Every continuous function defined on a compact interval is limited there and assumes a maximum and a minimum there. ${\displaystyle [a,b]\subset \mathbb {R} \;(a\leq b)}$

Or in detail:

(Ib) If there is a continuous function, then there are always arguments such that the inequality is satisfied for every other argument . ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle {\tilde {x}},{\hat {x}}\in [a,b]}$ ${\displaystyle x\in [a,b]}$ ${\displaystyle f({\hat {x}})\leq f(x)\leq f({\tilde {x}})}$

Or briefly and including the interim value theorem :

(II) For every continuous function there are arguments with . ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle {\tilde {x}},{\hat {x}}\in [a,b]}$ ${\displaystyle f([a,b])=[f({\hat {x}}),f({\tilde {x}})]}$

proof

Prerequisite : be a continuous function with and . ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle a\leq b}$

${\displaystyle M=f([a,b])}$be the set of all function values ​​that takes. ${\displaystyle f}$

The consequences and each hot inherent when a follower true for ever: . ${\displaystyle (y_{n}),y_{n}\in M}$ ${\displaystyle (x_{n}),x_{n}\in [a,b]}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle f(x_{n})=y_{n}}$

${\displaystyle (x_{k})}$or be a partial sequence resulting from a suitable selection from or , where . ${\displaystyle (y_{k})}$ ${\displaystyle (x_{n})}$ ${\displaystyle (y_{n})}$ ${\displaystyle k\in K\subset \mathbb {N} }$

A. Claim : Every sequence has a subsequence that converges to a. ${\displaystyle (y_{n})}$ ${\displaystyle (y_{k})}$ ${\displaystyle y_{l}\in M}$

Proof: The corresponding sequence is restricted because of . With the Bolzano-Weierstrass Theorem can be derived from a convergent subsequence select. Since is compact, it converges to a . Since in is continuous, the corresponding sequence converges to the result of the continuity test against . ${\displaystyle (y_{n})}$ ${\displaystyle (x_{n})}$ ${\displaystyle x_{n}\in [a,b]}$ ${\displaystyle (x_{n})}$ ${\displaystyle (x_{k})}$ ${\displaystyle [a,b]}$ ${\displaystyle (x_{k})}$ ${\displaystyle x_{l}\in [a,b]}$ ${\displaystyle f}$ ${\displaystyle [a,b]}$ ${\displaystyle (y_{k})}$ ${\displaystyle y_{l}=f(x_{l})\in M}$

B. Claim : is bounded above in [a, b] . ${\displaystyle f}$

The evidence is indirect . - Assumption : there is no upper limit. ${\displaystyle f}$

Then there is a strictly monotonically increasing and (definitely) divergent sequence . Every partial sequence of is also divergent. This is contradictory, because with A. a convergent partial sequence can be selected. ${\displaystyle (y_{n})}$ ${\displaystyle (y_{k})}$ ${\displaystyle (y_{n})}$ ${\displaystyle (y_{n})}$ ${\displaystyle (y_{k})}$

So there is an upper limit and a supremum . ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle s\in \mathbb {R} }$

C. Claim : assumes a maximum in [a, b]. ${\displaystyle f}$

A sequence can be created from suitably chosen elements of which converges against the supremum of . Any subsequence of also converges to . With A. there is a subsequence of that converges against . Because of the uniqueness of the limit value is the maximum of the assertion. ${\displaystyle M}$ ${\displaystyle (y_{n})}$ ${\displaystyle s}$ ${\displaystyle M}$ ${\displaystyle (y_{k})}$ ${\displaystyle (y_{n})}$ ${\displaystyle s}$ ${\displaystyle (y_{k})}$ ${\displaystyle (y_{n})}$ ${\displaystyle y_{l}=f(x_{l})\in M}$ ${\displaystyle s=y_{l}}$

D. Claim : is bounded downwards in [a, b] and assumes a minimum there. ${\displaystyle f}$

To prove this, replace "above" by "below", "rising" by "falling", "supremum" by "infimum" and "maximum" by "minimum" in B. and C.

Remarks

• The proposition is a pure existential proposition. He's not constructive. That means: It does not provide a method to actually determine the extreme points. For differentiable functions, the curve discussion methods can be used to determine the extremes of a function.
• The principle of minimum and maximum is in a certain sense characteristic of . Its unqualified validity is equivalent to the supremum axiom . ${\displaystyle \mathbb {R} }$

generalization

The same theorem - according to the versions (Ia) or (Ib) - also applies if any compact topological space is used instead of a compact real interval : Continuous images of compact topological spaces under real-valued functions are always closed within the real numbers and limited.

In fact, this statement can be generalized even further: The image of a compact topological space under a continuous function is again compact. Since compact subsets of metric spaces (especially of ) are always closed and bounded, the above statement follows immediately. ${\displaystyle \mathbb {R} }$

Since the images of connected topological spaces are connected again under continuous functions and the connected subsets of are precisely the intervals , version (II) also presents itself as a special case of a general topological situation. ${\displaystyle \mathbb {R} }$

Sources and background literature

• Otto Forster : Analysis 2 (=  basic course in mathematics ). 8th, updated edition. Vieweg + Teubner , Wiesbaden 2008, ISBN 978-3-8348-9541-7 .
• Horst Schubert : Topology. An introduction (=  mathematical guidelines ). 4th edition. BG Teubner Verlag , Stuttgart 1975, ISBN 3-519-12200-6 . MR0423277

Web links

Wikibooks: Math for Non-Freaks: Theorem of Minimum and Maximum  - Learning and teaching materials

Individual evidence

1. ↑ An example is the recursively defined sequence : any, any. ${\displaystyle (y_{n})}$ ${\displaystyle y_{0}}$ ${\displaystyle y_{n}+1\leq y_{n+1}}$
2. ↑ One example is the sequence recursively defined : arbitrarily . ${\displaystyle (y_{n})}$ ${\displaystyle y_{0}}$ ${\displaystyle s\geq y_{n+1}\geq (s+y_{n})/2}$
3. ↑ In the proof of the existence of the minimum there are examples of recursively defined sequences of the proof: in B .: any, any, or in C .: any, any. ${\displaystyle (y_{n})}$ ${\displaystyle y_{0}}$ ${\displaystyle y_{n}-1\geq y'_{n+1}}$ ${\displaystyle (y_{n})}$ ${\displaystyle y_{0}}$ ${\displaystyle s\leq y_{n+1}\leq (s+y_{n})/2}$
4. ↑ Horst Schubert: Topology. 1975, p. 62
5. ↑ The theorem of minimum and maximum can even be extended to the case of semi-continuous functions . See evidence archive .
6. ↑ There is another generalization that also includes the case of sequential compact spaces .