Answer
$h$ is discontinuous at $x=0.$
Work Step by Step
$x=0$ is of interest to us, as it is the only value where $h(x)$ can have a discontinuity.
Left-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 0^{-}}h(x)=0+2=2$
Right-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 0^{+}}f(x)=2(0)+2=2$
The limit exists, $L=2.$
Function value:$\quad h90)=0$
The limit does not equal the function value, so $h$ is discontinuous at $x=0.$
