Answer
$f(x)$ is neither left nor right continuous at $x=2$.
Work Step by Step
Given $$
f(x)=\left\{\begin{array}{ll}
{\dfrac{x^{2}-3 x+2}{|x-2|}} & {x \neq 2} \\
{0} & {x=2}
\end{array}\right.
$$
From the figure,
$$
\lim _{x \rightarrow 2^{-}} f(x) \neq \lim _{x \rightarrow 2^{-}} f(x) \neq f(2)
$$
Then $f(x)$ is neither left nor right continuous at $x=2$ (there is a jump discontinuity at $x=2$).
