aequationes mathematicae

, Volume 59, Issue 3, pp 248–254

# Bounded solutions of the Golab—Schinzel equation

• J. Brzdek

## Summary.

Let $${\Bbb K}$$ be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over $${\Bbb K}$$ n be a positive integer, and $$f : X \to {\Bbb K}$$ be a solution of the functional equation¶¶$$f(x + f(x)^n y) = f(x) f(y)$$.¶We prove that, if there is a real positive a such that the set $$\{ x \in X : |f(x)| \in (0, a)\}$$ contains a subset of second category and with the Baire property, then f is continuous or $$\{ x \in X : |f(x)| \in (0, a)\}$$ for every $$x \in X$$. As a consequence of this we obtain the following fact: Every Baire measurable solution $$f : X \to {\Bbb K}$$ of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set $$A \subset X$$ with $$f(X \backslash A) = \{ 0 \})$$.

## Keywords

Positive Integer Functional Equation Complex Number Bounded Solution Equal Zero
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.