Question

If $f(x+y)=f\left(x\right)f\left(y\right)$ for all $x,yϵR$ and $f\left(x\right)=1+g\left(x\right)G\left(x\right)$, where $\underset{x\to 0}{lim}g\left(x\right)=0$ and $\underset{x\to 0}{lim}G\left(x\right)$ exists, prove that $f\left(x\right)$ is continuous at all $xϵR$.

Answer 1

Put $y=0$ in the given equation $f(x+y)=f\left(x\right)f\left(y\right)$

we get, $f\left(x\right)=f\left(x\right)f\left(0\right)\dots \left(1\right)$

Also given $f\left(x\right)=1+g\left(x\right)G\left(x\right)$

$\Rightarrow f\left(0\right)=1+g\left(0\right)G\left(0\right)\dots \left(2\right)$

$\because \underset{x\to 0}{lim}g\left(x\right)=0$

$\therefore$ From eqn (2)

$f\left(0\right)=1+0=1$

Therefore $f\left(0\right)$ exists and from the eqn (1) $f\left(x\right)$ is a continous function.