Answer
$f(2)=6$ makes $f$ continuous at $x=2$.
Work Step by Step
The function $f$ is discontinuous at $x=0,$ where $\lim _{x \rightarrow 0-} f(x)=\infty$ and $\lim _{x \rightarrow 0+} f(x)=2 .$ The function $f$ is also discontinuous at $x=2,$ where $\lim _{x \rightarrow 2-} f(x)=6$ and $\lim _{x \rightarrow 2+} f(x)=6 .$ Because the two one-sided limits exist and are equal at $x=2,$ the discontinuity at $x=2$ is removable. Assigning $f(2)=6$ makes $f$ continuous at $x=2$.
