Answer
No discontinuities.
Work Step by Step
$x=1$ is of interest to us, as it is the only value where $g(x)$ can have a discontinuity.
Left-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 0^{-}}g(x)=1-1=0$
Right-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 0^{+}}g(x)=1-1=0$
They are equal $\Rightarrow$ a limit does exist at $x=1$, and it is $L=0.$
Now, the function value at $x=1$ is:$\quad g(1)=1-(1)=0.$
They are equal, so $g$ is continuous at $x=1$.
