Bernstein-Doetsch's theorem

The set of Amber Doetsch is a tenet of mathematical sub-region of Analysis , which to a work of the mathematician Felix Bernstein and Gustav Doetsch back from the year 1915th The theorem gives a sufficient condition under which certain convex functions of Euclidean space are already continuous .

Formulation of the sentence

Bernstein-Doetsch's theorem can be stated as follows:

Let be a convex and at the same time open subset of the . ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$

Let be a Jensen convex function , i.e. a real-valued function , which of the condition ${\displaystyle f\colon \Omega \to \mathbb {R} }$

${\displaystyle f\left({\frac {x+y}{2}}\right)\leq {\frac {f(x)+f(y)}{2}}}$

may suffice for all . ${\displaystyle x,y\in \Omega }$

Next there is at least one point in such a way that for a open environment , the restriction is limited to the top was. ${\displaystyle x_{0}\in \Omega }$ ${\displaystyle U(x_{0})\subseteq \Omega }$ ${\displaystyle f|_{U(x_{0})}}$

Then:

${\displaystyle f}$is continuous at every point . ${\displaystyle \Omega }$

Historical note

Already in 1906 Johan Ludwig Jensen delivered a forerunner result to the Bernstein-Doetsch theorem, by showing that the corresponding state of affairs holds for convex functions on open real intervals .

Inferences

Bernstein-Doetsch's theorem leads directly to the following corollary:

A Jensen convex function given on an open and convex subset of Euclidean space is either continuous or discontinuous at every point .

In addition, you win with the set of Amber Doetsch the following fundamental result that the Polish mathematician Marek Kuczma in his famous monograph An Introduction to the Theory of Functional Equations and Inequalities as The basic theorem titled. This says:

If is a real-valued function for a convex open subset des , then both Jensen-convex and continuous if and only if ${\displaystyle f\colon \Omega \to \mathbb {R} }$ ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle f}$

if for every two points and every real number always the inequality ${\displaystyle x,y\in \Omega }$ ${\displaystyle \lambda \in [0,1]}$

${\displaystyle f\left(\lambda x+(1-\lambda )y\right)\leq \lambda f(x)+(1-\lambda )f(y)}$

is satisfied.

The sentences of Sierpiński and Fréchet

A proposition goes back to the Polish mathematician Wacław Sierpiński , the question of which is similar to that of Bernstein-Doetsch's proposition, although its proof is based on different methods. It reads:

Given a convex open subset of and a Jensen convex function on it . ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle f\colon \Omega \to \mathbb {R} }$

Then:

If it is measurable , then it is already continuous. ${\displaystyle f}$ ${\displaystyle f}$

Sierpiński's theorem, in turn, leads directly to a theorem, which was formulated for the case of dimension by the French mathematician Maurice Fréchet in 1913: ${\displaystyle n=1}$

Every measurable additive function is continuous. ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$

Related result for standardized spaces

There is a related result to the Bernstein-Doetsch theorem, which treats the case of convex real-valued functions on normalized spaces . It can be formulated as follows:

Let a normalized - vector space be given and in it a convex open subset and a convex real-valued function . ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle \Omega \subseteq X}$ ${\displaystyle f\colon \Omega \to \mathbb {R} }$

Then the following statements are equivalent:

(a) is continuous. ${\displaystyle f}$

(b) is above semi-continuous . ${\displaystyle f}$

(c) There is a nonempty open subset such that it is bounded above. ${\displaystyle U_{0}\subseteq \Omega }$ ${\displaystyle f|_{U_{0}}}$

(d) There is at least one point where is continuous. ${\displaystyle x_{0}\in \Omega }$ ${\displaystyle f}$

If, in addition, is a Banach space , the following are even equivalent: ${\displaystyle X}$

(a ') is continuous. ${\displaystyle f}$

(b ') is above semi-continuous. ${\displaystyle f}$

(c ') is sub-semi-continuous . ${\displaystyle f}$

literature

• F. Bernstein and G. Doetsch : On the theory of convex functions . In: Mathematical Annals . tape 76 , 1915, pp. 514-526 , doi : 10.1007 / BF01458222 ( MR1511840 ).
• Maurice Fréchet: Pri la fukncia equacio f (x + y) = f (x) + f (y) . In: Enseignement Math. Band 15 , 1913, pp. 390-393 .
• Maurice Fréchet: A propos d'un article sur l'équation fonctionelle f (x + y) = f (x) + f (y) . In: Enseignement Math. Band 16 , 1914, pp. 136 .
• JLWV Jensen : Sur les fonctions convexes et les inégalités entre les valeurs moyennes . In: Acta Mathematica . tape 30 , 1906, pp. 175-193 ( MR1555027 ).
• Peter Kosmol : Optimization and Approximation (=  De Gruyter Studies ). 2nd Edition. Walter de Gruyter & Co. , Berlin 2010, ISBN 978-3-11-021814-5 ( MR2599674 ).
• Marek Kuczma : An Introduction to the Theory of Functional Equations and Inequalities . Cauchy's Equation and Jensen's Inequality. 2nd Edition. Birkhäuser Verlag , Basel 2009, ISBN 978-3-7643-8748-8 ( MR2467621 ).
• Wacław Sierpiński: Sur unproblemème concernant les ensembles mesurables superficiellement . In: Fundamenta Mathematicae . tape 1 , 1920, p. 112-115 ( -> web link ).
• Wacław Sierpiński: Sur les fonctions convexes mesurables . In: Fundamenta Mathematicae . tape 1 , 1920, p. 125–128 ( -> web link ).

Individual evidence

1. ↑ ^{a b} F. Bernstein, G. Doetsch: On the theory of convex functions. in: Math. Ann. 76, pp. 514-526
2. ^ ^{A b} Marek Kuczma: An Introduction to the Theory of Functional Equations and Inequalities. 2009, p. 155 ff
3. ↑ Bernstein / Doetsch, op.cit., P. 514
4. ↑ Kuczma. op.cit., p. 158
5. ↑ Kuczma. op. cit., pp. 161-162
6. ↑ ^{a b} Kuczma. op.cit., p. 241 ff
7. ↑ W. Sierpiński: Sur unproblemème concernant les ensembles mesurables superficiellement. in: Fund. Math. 1, pp. 112-115
8. ↑ Sierpiński, op. Cit., Pp. 125–128
9. ↑ Peter Kosmol: Optimization and Approximation. 2010, p. 328 ff., P. 331 ff.