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

Marek Kuczma was once born in 1935 in Katowice, Poland, and died there in 1991.

After completing highschool in his domestic city, he studied on the Jagiellonian college in Kraków. He defended his doctoral dissertation less than the supervision of Stanislaw Golab. within the 12 months of his habilitation, in 1963, he acquired a place on the Katowice department of the Jagiellonian collage (now collage of Silesia, Katowice), and labored there until eventually his death.

Besides his numerous administrative positions and his notable educating task, he entire very good and wealthy medical paintings publishing 3 monographs and one hundred eighty clinical papers.

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

"The moment 1/2 the identify of this ebook describes its contents thoroughly. most likely even the main dedicated professional wouldn't have inspiration that approximately three hundred pages could be written on the subject of the Cauchy equation (and on a few heavily comparable equations and inequalities). And the ebook is not at all chatty, and doesn't even declare completeness. half I lists the necessary initial wisdom in set and degree concept, topology and algebra. half II supplies information on options of the Cauchy equation and of the Jensen inequality [...], particularly on non-stop convex features, Hamel bases, on inequalities following from the Jensen inequality [...]. half III bargains with comparable equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex capabilities of upper order, subadditive features and balance theorems). It concludes with an day trip into the sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This publication is a true vacation for all of the mathematicians independently in their strict speciality. you'll think what deliciousness represents this publication for sensible 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**

**Sample text**

Now, H ∪ R ∪ H = X, and H ∪ R ∪ G = R ∪ G , since H = int G ⊂ G . Hence (H ∪C)\D = (R∪G )∩(P ∪R) = (R∩P )∪R∪(G ∩P )∪(G ∩R) = (G ∩P )∪R = A since R ∩ P ⊂ R and G ∩ R ⊂ R. 2). Now, the set H is open, D ⊂ P is of the ﬁrst category, and G \ H is a closed set without inner points, and so it is nowhere dense, and hence of the ﬁrst category. 2) that A has the Baire property. 2. If the space X is separable6 , then for every set A ⊂ X there exists a set B with the Baire property such that A ⊂ B and for every set Z with the Baire property containing A the set B \ Z is of the ﬁrst category.

Nm . nm = B. kj l ∈ Σ , l=1 since Σ is a σ-algebra. kj l ∈ Σ . nm \ z m=0 Let [. nm l . nm l . Then ∞ [. nm \ ∞ ∞ [. ] = m=0 z [. ] . 2) B \ A ∈ Σ. Since A ⊂ B, we have A = B \ (B \ A) ∈ Σ. 2. Every analytic set has the Baire property. In Part II of the present book we will encounter many sets without the Baire property, and hence non-analytic. 8. 8 Cantor-Bendixson theorem The Cantor-Bendixson theorem belongs to standard part of any university course on the topology of metric spaces. But we prove it here because of its application in the theory of analytic sets.

Take a Z ∈ Aα . , for every n ∈ N there exists a ξn < α such ξ<α that En ∈ Mξn . Hence, by (∗∗), En ∈ Aξn , n ∈ N, and ∞ En ∈ Z = n=1 Aξ δ ξ<α The proof of the converse implication is analogous. = Mα . 3. Borel sets 27 (iii) This results from the fact that B(X) is a σ-algebra (again a proof by transﬁnite induction is necessary, which is left to the reader). 2. We have Aα = α<Ω Mα = B(X) . α<Ω Proof. 1 (i) Aα ⊂ Mα+1 ⊂ Mα , and Aα ⊂ Mα . Similarly, for every α < Ω we have α<Ω α<Ω Mα ⊂ Aα+1 ⊂ α<Ω Aα , and α<Ω Mα ⊂ α<Ω Aα .