Bounded solutions of the Golab—Schinzel equation

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 $f(X \backslash A) = \{ 0 \})$.