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Introduction

  • Donald H. Hyers
  • George Isac
  • Themistocles M. Rassias
Chapter
  • 246 Downloads
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)

Abstract

The most famous functional equation is the Cauchy equation
$$ f\left( {x + y} \right) = f\left( x \right) + f\left( y \right)$$
(1.1)
any solution of which is called additive. It is known that, for real valued functions defined on the real line, every Lebesgue measurable solution of (1.1) is of the form f (x) =cx for some constant c. Nonmeasurable solutions also exist but they are “wild”, being discontinuous everywhere and unbounded on every interval. A discussion of these facts may be found in Chapter 2 of the book by J. Aczél and J. Dhombres (1989).

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Donald H. Hyers
  • George Isac
    • 1
  • Themistocles M. Rassias
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

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