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By Marek Kuczma

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 Krak?w. He defended his doctoral dissertation less than the supervision of Stanislaw Golab. within the yr of his habilitation, in 1963, he got 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 remarkable instructing job, he finished first-class and wealthy clinical paintings publishing 3 monographs and one hundred eighty medical papers.

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

"The moment half the identify of this publication describes its contents competently. most likely even the main committed expert wouldn't have suggestion that approximately three hundred pages will be written with reference to the Cauchy equation (and on a few heavily similar equations and inequalities). And the e-book is certainly not chatty, and doesn't even declare completeness. half I lists the mandatory initial wisdom in set and degree thought, topology and algebra. half II supplies information on strategies 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 offers with comparable equations and inequalities (in specific, Pexider, Hossz?, and conditional equations, derivations, convex features of upper order, subadditive features 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. possible think what deliciousness represents this publication for useful equationists." (B. Crstici, Zentralblatt f?r Mathematik)

 

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Extra info for An introduction to the theory of functional equations and inequalities

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Finally we give here two characterizations of the class A(X). 7. Let X be a complete and separable metric space. A set A ⊂ X is analytic if and only if there exist a Borel set B ⊂ X and a continuous function f : B → X such that f (B) = A. 2. 8. Let X be a complete and separable metric space. The family A(X) is the smallest class Z of subsets of X which fulfils the following three conditions (i) F (X) ⊂ Z; (ii) If Ai ∈ Z for i ∈ N, then also ∞ i=1 Ai and ∞ Ai ∈ Z; i=1 (iii) If Z ∈ Z, and f : Z → X is a continuous function, then f (Z) ∈ Z.

This formula defines a choice function w : A → A. P. J. Cohen [48] proved that the Axiom of Choice is independent of the remaining Axioms 1–8. A detailed discussion of The Axiom of Choice is found in Jech [154]. Exercises 1. Prove that for any two sets A, B there exists the set A ∩ B containing those and only those elements which are common to A and B. 2. The ordered pair a, b is defined as {a, {a, b}}, where {a, b} is the unordered pair consisting of a and b. Prove that a, b = c, d if and only if a = c and b = d.

5. Analytic sets 33 function f is continuous. To see this, take a sequence yk = (zk , nk ) ∈ z × N , k ∈ N, convergent to a y0 = (z0 , n0 ) ∈ z × N: lim yk = y0 . , nk = n0 for k sufficiently large. For such k k→∞ k→∞ we have f (yk ) = f (zk , n0 ) = fn0 (zk ) → fn0 (z0 ) = f (z0 , n0 ) = f (y0 ) , since fn0 is continuous. 4 is an analytic set. Moreover we have ∞ f (z × N) = Ai . , x = fn (z), whence x ∈ fn (z) = An and x ∈ ∞ Ai . , there exists a z ∈ z such that x = fn (z) = f (z, n) ∈ f (z × N).

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