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 college in Kraków. He defended his doctoral dissertation less than the supervision of Stanislaw Golab. within the yr of his habilitation, in 1963, he bought a place on the Katowice department of the Jagiellonian collage (now collage of Silesia, Katowice), and labored there until eventually his death.

Besides his a number of administrative positions and his amazing instructing task, he entire very good 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 practical equations and inequalities.

"The moment 1/2 the name of this booklet describes its contents correctly. most likely even the main dedicated professional do not need proposal that approximately three hundred pages may be written on the subject of the Cauchy equation (and on a few heavily similar equations and inequalities). And the ebook is in no way 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 options of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex features, 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 features of upper order, subadditive services and balance theorems). It concludes with an expedition into the sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

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

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

**Sample text**

Let X, Y be topological spaces, and f : X → Y a homeomorphism. Prove that for any A ⊂ X cl f (A) = f (cl A) , int f (A) = f (int A) , and hence the sets A and f (A) are both of the same category. 3. Let X, Y be topological spaces, X compact, and let f : X → Y be a one-to-one continuous function. Prove that f is a homeomorphism. 4. Prove that any subset of a ﬁrst category set is of the ﬁrst category, and the union of a ﬁnite or countable collection of ﬁrst category sets is of the ﬁrst category. 5.

Fix an x = {xi } ∈ X and consider the set Y ⊂ X consisting of the points {y1 , x2 , x3 , . } , {y1 , y2 , x3 , . } , {y1 , y2 , y3 , . 5) ............... , yi ∈ Yi . The cardinality of Y equals that of ∞ ∞ × Yi , and thus Y is countable. Take n=1 i=1 an x = {xi } ∈ X. 6) lim yni = xi . 1 that lim yn = x . n→∞ Thus Y is dense in X, and consequently X is separable. 2. If all spaces Xi are complete, then so is also X. Proof. Let {xn } ⊂ X , xn = {xni } , be an arbitrary Cauchy sequence in X. 1) (xn , xn+k ) 1 i (xni , xn+k,i ) 2i 1 + i (xni , xn+k,i) 30 Chapter 2.

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.