Answer
No discontinuities.
Work Step by Step
$x=0$ and $x=2$ are of interest to us, as they are the only values where $g(x)$ can have a discontinuity.
At $x=0$:
Left-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 0^{-}}g(x)=0+2=2$
Right-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 0^{+}}g(x)=2(0)+2=2$
The limit exists, $L=2.$
Function value:$\quad g(0)=2(0)+2=2$
Thus, $g$ is continuous at 0.
At $x=2$:
Left-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 2^{-}}g(x)=2(2)+2=6$
Right-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 2^{+}}g(x)=2^{2}+2=6$
The limit exists, $L=6.$
Function value:$\quad g(0)=2^{2}+2=6$
Thus, $g$ is continuous at $2$.
So we have no discontinuities.
