Answer
For $g(x)$ to be continuous at the origin, we only need to extend $g(x)$ to cover the point $(0,1)$.
Work Step by Step
$$g(x)=\frac{\tan(\tan x)}{\tan x}$$
1) First, find $\lim_{x\to0}\frac{\tan(\tan x)}{\tan x}$
$$A=\lim_{x\to0}\frac{\tan(\tan x)}{\tan x}$$
Take $w=\tan x$. Then as $x\to0$, we would have $w=\tan x\to\tan0=0$
$$A=\lim_{w\to0}\frac{\tan w}{w}$$ $$A=\lim_{w\to0}\frac{\sin w}{w\cos w}$$ $$A=\lim_{w\to0}\frac{\sin w}{w}\times\lim_{w\to0}\frac{1}{\cos w}$$ $$A=1\times\frac{1}{\cos0}$$ $$A=\frac{1}{1}=1$$
2) So here we see that $g(x)$ is not defined as $\tan x=0$ or $x=0$; in other words, $g(x)$ is not continuous at $x=0$.
Yet we already have $\lim_{x\to0}\frac{\tan(\tan x)}{\tan x}=1$, which means $g(x)$ approaches $1$ as $x\to0$.
Therefore, for $g(x)$ to be continuous at the origin, we only need to extend $g(x)$ to cover the point $(0,1)$.
