Theorems and Counterexamples in Mathematics by Bernard R. Gelbaum

By Bernard R. Gelbaum

These counterexamples, prepared in response to trouble or sophistication, deal ordinarily with the a part of research referred to as "real variables," beginning on the point of calculus. the 1st half the e-book issues features of a true variable; issues comprise the genuine quantity process, features and boundaries, differentiation, Riemann integration, sequences, limitless sequence, uniform convergence, and units and degree at the actual axis. the second one part, encompassing larger dimensions, examines features of 2 variables, aircraft units, quarter, metric and topological areas, and serve as areas. This quantity comprises a lot that might end up appropriate for college students who've now not but accomplished a primary direction in calculus, and plentiful fabric of curiosity to extra complex scholars of research in addition to graduate scholars. 12 figures. Bibliography. Index. Errata.

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3] W. Fechner, On some inequalities connected with the exponential function, Arch. Math. (Brno) 44 (2008), No. 3, 217–222. [4] W. Fechner, A functional characterization of two inequalities between means, (submitted). [5] Z. Gajda, On stability of the Cauchy equation on semigroups, Aequationes Math. 36 (1988), No. 1, 76–79. [6] Z. Gajda, Invariant means and representations of semigroups in the theory of func´ ¸ skiego w Katowicach [Scientific tional equations, Prace Naukowe Uniwersytetu Sla ´ ¸ ski, Katowice, 1992.

Y−x (16) Then the estimate (15) holds true. Proof. It suffices to apply Proposition 2, then [3, Theorem 2], Remark 1 and the Hahn– Banach Theorem. One may obtain analogous results for equation (13). It suffices to apply [4, Theorem 7] and repeat the reasoning used previously. Proposition 3. Given an ε ≥ 0, let f0 : I → R satisfy f0 (y) − f0 (x) ≤ 4f0 y−x 6 x+y 2 + f0 (x) + f0 (y) + ε, for each x, y ∈ I such that x < y. Then f0 (x) = f (x) − 61 ε for x ∈ I, where f is a solution of (5). Theorem 4.

Capture Conditions in a Pursuit–Evasion Game between Players with Biproper Dynamics, Journal of Optimization Theory and Applications , 126:3, 503– 528, 2005. , A Folk Theorem for Repeated Sequential Games, The Review of Economic Studies, Volume 69:2 493–512(20), 2002. In: Functional Equations, Difference Inequalities... ISBN 978-1-60876-461-7 c 2010 Nova Science Publishers, Inc. Editor: John Michael Rassias, pp. 37-46 Chapter 3 S OME S TABILITY R ESULTS FOR E QUATIONS AND I NEQUALITIES C ONNECTED WITH THE E XPONENTIAL F UNCTION Włodzimierz Fechner and Roman Ger∗ Institute of Mathematics, Silesian University Bankowa 14, 40-007 Katowice, Poland Abstract We generalize some earlier results connected with the Hyers-Ulam stability of functional equations and inequalities related to the exponential function.