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Let $f: \mathbb R\rightarrow \mathbb R$ a derivable function but not zero, such that $f'(0) = 2$ and $$ f(x+y)= f(x)\cdot \ f(y)$$ for all $x$ and $y$ belongs $\mathbb R$. Find $f$.

My first answer is $f(x) = e^{2x}$, and I proved that there are not more functions like $f(x) = a^{bx}$ by Existence-Unity Theorem (ODE), but I don't know if I finished.

What do you think about this sketch of proof's idea?

Thanks,

I'll be asking more things.

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    See http://math.stackexchange.com/questions/22069/is-there-a-name-for-function-with-the-exponential-property-fxy-fx-times-f2012-07-26

2 Answers 2