For Problems $1$ and $2,$ consider the following graph of a function $f(x)$ :

$(1)$ Calculate the following limits $($if a limit does not exist, explain why$)$:

$($a$)\underset{x-2^{-}}{lim}f(x)$

$($b$)\underset{x-2}{lim}f(x)$

$($c$)\underset{x{0}^{-}}{lim}f(x)$

$($d$)\underset{x0}{lim}f(x)$

$($e$)f(0)$

$($f$)\underset{x3}{lim}f(x)$

$($g$)f(5)$

$($h$)\underset{x{5}^{+}}{lim}f(x)$

$(2)($a$)$ Find all points where the function above is not continuous, Explain why the function is not continuous there.

$($b$)$ For which of those points is the discontinuity removable? Explain how to remove it$.$