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Zeitschrift für Analysis und ihre Anwendungen


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Volume 3, Issue 6, 1984, pp. 549–554
DOI: 10.4171/ZAA/129

Published online: 1984-12-31

Redundancy Conditions for the Functional Equation $f(x + h(x)) = f(x) + f(h(x))$

Gian Luigi Forti[1]

(1) Università di Milano, Italy

Consider the functional equation $f(x+ h(x)) = f(x) + f(h(x))$, where $h: \mathbb R \to \mathbb R$ is a given continuous function, $h(0) = 0$. It is proved if the set of all zeros of $h$ and of all points where $h(x) = -x$ is not "too much dense", then the continuous and at $x = 0$ differentiable solution $f: \mathbb R \to \mathbb R$ of the functional equation under consideration is $f(x) = xf’(0)$ for all real $x$.

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Forti Gian Luigi: Redundancy Conditions for the Functional Equation $f(x + h(x)) = f(x) + f(h(x))$. Z. Anal. Anwend. 3 (1984), 549-554. doi: 10.4171/ZAA/129