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I am looking at Pexider's equation $f(x+y)=g(x)+h(y)$, where $f,g,h$ are continuous functions but are defined over bounded domains. Specifically, $f,g,h$ each is defined on a real interval (of length $>0$), but not necessarily the entire real line. (The domains are such that the functional equation holds throughout the respective domains.) It seems that uniqueness of the Pexider's solution still holds. Is this true? any references?

If necessary one may assume:

  • all three domains contain $0$ in their interior.
  • $f$ is strictly monotone.

Thanks.

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First, you can do substitutions $f(x) = g(x) + h(0)$ to reduce the question to a Cauchy functional equation.

The resulting functional equation $\bar h(x + y) = \bar h(x)+\bar h(y)$, $(x,y)\in (-a,b)\times(-c,d)$ can be extended to the entire real line. Notice that $x+y$ will lie outside the original domain for sufficiently small/large $x$ and $y$. Afterwards, you can simply use the standard Cauchy solution.

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