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An application of nonstandard analysis to functional equations. (English) Zbl 0576.39008
The author shows that all measurable solutions f of the functional equation \(f(x+y)=g(f(x),f(y),x,y)\) where g is continuous, are continuous. A special case is the Cauchy functional equation \(f(x+y)=f(x)+f(y)\). The methodology is nonstandard analysis, (including Loeb measure).
MSC:
39B99 Functional equations and inequalities
03H05 Nonstandard models in mathematics
26E35 Nonstandard analysis
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