Answer
$f$ is discontinuous at $x=0$.
Work Step by Step
$x=1$ and $x=2$ are of interest to us, as they are the only values where $f(x)$ can have a discontinuity.
At $x=0$:
Left-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 0^{-}}f(x)=1/x|_{x\rightarrow 0}=\infty$
The limit at $x=0$ does not exist, so $f$ is discontinuous at $0$.
At $x=2$:
Left-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 2^{-}}f(x)=2$
Right-sided limit:$\qquad \displaystyle \lim_{x\rightarrow 2^{+}}f(x)=2^{2-1}=2$
The limit exists, $L=2.$
Function value:$\quad f(2)=2$
Thus, $f$ is continuous at $2$.
