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On the behavior of mappings which do not satisfy Hyers-Ulam stability. (English) Zbl 0761.47004

The main result of the paper is the following

Theorem. There exists a continuous function f:RR, satisfying

|f(x+y)-f(x)-f(y)||x|+|y|,

for any x,yR, with lim x (f(x)/x)=.

This theorem gives an example to show that a stability theorem of Hyers- Rassias-Gajda-Ulam cannot be proved for p=1.


MSC:
47A58Operator approximation theory
41A35Approximation by operators (in particular, by integral operators)
47J05Equations involving nonlinear operators (general)