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 college in Kraków. He defended his doctoral dissertation lower than the supervision of Stanislaw Golab. within the 12 months 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 his death.

Besides his numerous administrative positions and his amazing educating job, he finished first-class and wealthy clinical 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 practical equations and inequalities.

"The moment half the identify of this booklet describes its contents properly. most likely even the main committed professional wouldn't have idea that approximately three hundred pages will be written almost about the Cauchy equation (and on a few heavily comparable equations and inequalities). And the booklet is certainly not 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 info on recommendations of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex services, 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 expedition into the sphere of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This ebook is a true vacation for the entire mathematicians independently in their strict speciality. you'll think what deliciousness represents this e-book 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**

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Let Y be an arbitrary space. For every set A ⊂ X × Y and every x0 ∈ X we write A[x0 ] = {y ∈ Y | (x0 , y) ∈ A} . 2. Let Y be a separable5 topological space, and let A ⊂ X ×Y be a nowhere dense set. 5) is nowhere dense. , Kuratowski [196]). Since in the sequel the results of the present chapter will be used for X = RN , which is separable, we restrict ourselves to separable X only. Separable means here having an at most countable neighbourhood base. 8, the assumption that Y is separable is essential.

Proof. Let X be a complete and separable metric space. 7) where K(x, r) is the open ball centered at x with the radius r. Denoting by d(A) the 1 for every x ∈ X. 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.

1. Outer and inner measure 51 Proof. For every n ∈ N take a real number an < mi (An ) and a closed set Fn ⊂ An such that m(Fn ) > an . 15) k Take an arbitrary k ∈ N. The set k Fn ⊂ A. 16) n=1 since the sets Fn are disjoint, just like An . Now let a1 → mi (A1 ), . . , ak → mi (Ak ). 16) k mi (A) mi (An ) . 13). 13). 13) A1 = B , An = ∅ for n the inner measure: If B ⊂ A, then mi (B) 2, we obtain the monotonicity of mi (A) . 7. If An ⊂ RN , n ∈ N, are pairwise disjoint measurable sets, and A ⊂ RN is arbitrary, then ∞ mi A ∩ ∞ mi (A ∩ An ) .