## 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## Preview

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