Answer
a.
f is discontinuous at $-4,-2,2,$ and $4$
b.
At $-4$, from neither side
At $-2$, from the left
At $2$, from the right
At $4$, from the right
Work Step by Step
A function $f$ is continuous at a number $a$ if $\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$.
A function $f$ is continuous from the right at a number $a$ if
$\displaystyle \lim_{x\rightarrow a^{+}}f(x)=f(a)$
and
$f$ is continuous from the left at $a$ if
$\displaystyle \lim_{x\rightarrow a^{-}}f(x)=f(a)$
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a.
$f(-4)$ is not defined,
At $x=-2,\ 2$, and $4$,
the left and right limits are not equal (so the limits do not exist).
b.
At $-4$, $f$ is not continuous from either side since $f(-4)$ is not defined.
At $-2$, $f$ is continuous from the left since $\displaystyle \lim_{x\rightarrow-2^{-}}f(x)=f(-2)$.
At $2,\ f$ is continuous from the right since $\displaystyle \lim_{x\rightarrow 2^{+}}f(x)=f(2)$.
At $4,\ f$ is continuous from the right since $\displaystyle \lim_{x\rightarrow 4^{+}}f(x)=f(4)$
