A function  $f:R\to R$  satisfies the equation $f(x+y)=f\left(x\right).f\left(y\right)$  for all x,y in R and  $f\left(x\right)\ne 0$  for any $x\in R$  . Let the function be differentiable at x=0 and  $f\text{'}\left(0\right)=2$ then.

Clearly, for x=y=0; f(0) = 1
 $f\text{'}\left(x\right)=\underset{k\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}=2.f9x$
Integrating  $f\left(x\right)=e^{2x}$  from this all the remaining follows.
$\Rightarrow e^{2x}$   is every  where continuous
 $\Rightarrow f\left(\frac{1}{2}\right)=e\to$  an irrational number
Hence, (A),(B) (C) and (D) are correct.