A function $f$ is defined in $R$, and $f'(0)$ exist.
Let $f(x+y)=f(x)f(y)$ then prove that $f'$ exists for all $x$ in $R$.
I find this problem in here, and wonder the proof stated below has any problems?
Since $f'(0)=\lim_{h \rightarrow 0} \frac{f(0)(f(h)-1)}{h}=f(0)\lim_{h \rightarrow 0} \frac{f(h)-1}{h}$
it means $\lim_{h \rightarrow 0} \frac{f(h)-1}{h}$ exists.
$f'(x)==\lim_{h \rightarrow 0} \frac{f(x)(f(h)-1)}{h}=f(x)\lim_{h \rightarrow 0} \frac{f(h)-1}{h}$
thus $f'(x)$ exists if $f(x)$ exists.
But we can differentiate a function if it has a function value $f(x)$, thus $f'$ exists for all $x$ in $R$.
I know it's not a perfect one,
just want to know it can be another way to solve that problem or not.
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$f(0)=0$ case:
If $f(0)=0$, because $f(x+y)=f(x)f(y)$,
$f(x)=f(x+0)=f(x)f(0)=0$ thus $f'(x)=f'(0)$. Is that right?