$ \forall $ $x,y \in \mathbb{R}$ We know that the most trivial functions satisfying the functional equation
$$f(x+y)=f(x)f(y)$$
as $f(x)=a^x$, $f(x)=0$ and $f(x)=1$.
Are there any other functions
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1Is $f$ continuous? – Anurag A Jul 11 '17 at 10:57
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2Duplicate of https://math.stackexchange.com/questions/38376/classifying-functions-of-the-form-fxy-fxfy. – lhf Jul 11 '17 at 10:58
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@anurag $f$ need not be continous – Ekaveera Gouribhatla Jul 11 '17 at 11:01
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Possible duplicate of Find all functions $f$, defined over real numbers that satisfy $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$ – Simply Beautiful Art Jul 11 '17 at 12:00
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@SimplyBeautifulArt: The only solution to that is the trivial one, which is explicitly excluded in this question. – user21820 Jul 11 '17 at 12:27
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Oop, my bad then @user21820 – Simply Beautiful Art Jul 11 '17 at 12:27
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That is not an easy question.
Consider a function $g\colon\mathbb{R}\longrightarrow\mathbb R$ such that $(\forall x,y\in\mathbb{R}):g(x+y)=g(x)+g(y)$ and define $f=e^g$. Then, clearly, $(\forall x,y\in\mathbb{R}):f(x+y)=f(x)f(y)$. So, are there functions like $g$ other than those defined by $g(x)=cx$, for some real $c$? Well, it's complicated. Basically, it depends upon which set theory you are working with.
José Carlos Santos
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