# An Introduction to the Theory of Functional Equations and by Marek Kuczma (auth.), Attila Gilányi (eds.)

By Marek Kuczma (auth.), Attila Gilányi (eds.)

Marek Kuczma used to be born in 1935 in Katowice, Poland, and died there in 1991.

After completing highschool in his domestic city, he studied on the Jagiellonian collage in Kraków. He defended his doctoral dissertation below the supervision of Stanislaw Golab. within the yr of his habilitation, in 1963, he received a place on the Katowice department of the Jagiellonian college (now college of Silesia, Katowice), and labored there until eventually his death.

Besides his a number of administrative positions and his impressive instructing task, he comprehensive first-class and wealthy medical paintings publishing 3 monographs and a hundred and eighty clinical papers.

He is taken into account to be the founding father of the prestigious Polish university of useful equations and inequalities.

"The moment half the name of this publication describes its contents safely. most likely even the main dedicated expert shouldn't have proposal that approximately three hundred pages might be written with regards to the Cauchy equation (and on a few heavily comparable equations and inequalities). And the ebook is on no account chatty, and doesn't even declare completeness. half I lists the mandatory initial wisdom in set and degree idea, topology and algebra. half II supplies information on recommendations of the Cauchy equation and of the Jensen inequality [...], particularly on non-stop convex services, Hamel bases, on inequalities following from the Jensen inequality [...]. half III bargains with similar equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex services of upper order, subadditive services and balance theorems). It concludes with an day trip into the sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This booklet is a true vacation for all of the mathematicians independently in their strict speciality. you will think what deliciousness represents this publication for useful equationists." (B. Crstici, Zentralblatt für Mathematik)

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**Extra info for An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality**

**Example text**

1) m(A) = m(A ) = 0, whence m(RN ) = m(A ∪ A ) = m(A) + m(A ) = 0, a contradiction. 1. Let A ⊂ RN be an arbitrary set. Then the following conditions are equivalent: (i) mi (A ) = 0; (ii) for every measurable set E of positive measure we have A ∩ E = ∅; (iii) for every measurable set E we have me (A ∩ E) = m(E); (iv) for every open interval3 I ⊂ RN we have me (A ∩ I) = m(I). Proof. First we prove that condition (i) implies (ii). Let a set A ⊂ RN fulﬁl (i), and let E ∈ L be an arbitrary set such that m(E) > 0.

7) a countable cover: ∞ Kn(1) , d Kn(1) X= 1, n=1 and also ∞ cl Kn(1) , d cl Kn(1) X= 1. nm } , m, n1 , . . , nm ∈ N, as follows. We put (1) Dn = cl Kn . nm for all n1 , . . nm 1 2m . 4. nm ∩ cl Kn(m) , nm+1 ∈ N . nm } , m, n1 , . . nm , m, n1 , . . , nm , nm+1 ∈ N. nm . 9) deﬁnes a function f : z → X. nm , n1 . . nm ∈ N, form a cover of X. nm . The sequence {ni } represents a point z ∈ z and we have f (z) = x. Thus f is onto: f (z) = X . It remains to show that f is continuous. Fix an ε > 0 and choose a p ∈ N so that 1 1 < ε.

But we prove it here because of its application in the theory of analytic sets. Let X be a metric space and A ⊂ X. A point x ∈ X is called a point of accumulation of A iﬀ for every neighbourhood U of x we have card (U ∩ A) > 1. The set of all the accumulation points of A is denoted by Ad . A point x ∈ A is called a point of condensation of A iﬀ for every neighbourhood U of x we have card (U ∩ A) > ℵ0 . The set of all the condensation points of A is denoted by A• . A set A ⊂ X is called perfect iﬀ A = Ad .