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I am looking for the general continuous solutions $f:D \rightarrow \mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=\{x \in\mathbb C: |x|<1\}$. (Complex to reals would also help).

I tried writing out the functional equation in matrix representation and linking it to solutions in Aczel's textbook. However, the most I got out of this approach was that $f$ is a group homomorphism (well, yes...). The chapter on Cauchy's equation for complex numbers does not seem to address this problem at all. Related questions such as this or this did not clarify my question either.

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