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 less 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 college (now collage of Silesia, Katowice), and labored there until his death.

Besides his numerous administrative positions and his amazing educating task, he complete very good and wealthy clinical paintings publishing 3 monographs and a hundred and eighty medical papers.

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

"The moment half the identify of this e-book describes its contents correctly. most likely even the main committed expert shouldn't have inspiration that approximately three hundred pages may be written almost about 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 recommendations of the Cauchy equation and of the Jensen inequality [...], particularly on non-stop convex capabilities, Hamel bases, on inequalities following from the Jensen inequality [...]. half III offers 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 sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This ebook 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)

**Read Online or Download An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality PDF**

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

**Example text**

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. Put δ = p+1 . Take z , z ∈ z, z = {ni }, z = {ni }. (z , z ) < δ means 2p−1 2 ∞ i=1 1 1 |ni − ni | < p+1 . 10) Since ni and ni are integers, ni = ni implies |ni − ni | 1 + |ni − ni | 1 .

1) that me (∅) = mi (∅) = 0 . 2. If A ∈ L, then me (A) = mi (A) = m(A) . 2) Proof. 1 there exist, for every ε > 0, a closed set F and open sets G, U, V such that F ⊂ A ⊂ G, G \ A ⊂ U , A\ F ⊂ V , m(U ) < ε , m(V ) < ε . , m(G) − ε m(A) m(F ) + ε . 3) shows that m(A) = inf m(G) = G⊃A G open sup m(F ) . 2). 3. 2) holds. Proof. 4) implies that there exist, for every ε > 0, a closed set F and an open set G such that F ⊂ A ⊂ G and ε ε me (A) = mi (A) m(F ) + . m(G) − 2 2 Hence m(G \ F ) = m(G) − m(F ) ε, and clearly the set G \ F is open, G \ A ⊂ G \ F , A \ F ⊂ G \ F .

Clearly B(X) has properties (i) and (ii), so K ⊂ B(X) . To prove the converse inclusion, we will show that for every α < Ω Aα ∪ Mα ⊂ K . 1) 28 Chapter 2. 1) is true by (i) and by the fact that G(X) ⊂ Fσ ⊂ K. Assume that for a certain α < Ω we have Aξ ∩ Mξ ⊂ K for every ξ < α. Then also Aξ ∪ ξ<α Mξ ⊂ K , ξ<α and by (ii) Aξ δ ∪ ξ<α Mξ σ ⊂ K. 1) holds. 2. 4. If the space X is separable, then card B(X) c. Proof. In a separable space X we have card G(X) c (since every open set may be represented as a union of base neighbourhoods, and there exists a countable base), and hence also card F (X) c.