a) In order for to be continuous at , we need to have
By definition of , we know . Meanwhile, the limits are
so is indeed continuous at .
b) We use the first derivative test (FDT) here, but when we compute the derivative of a piecewise function, we have to be careful at the points where the pieces "split off", because it's possible that the derivative does not exist at these points, yet an extreme value can still occur there. (Consider, for example, at .)
In this case,
We find the critical points for each piece over their respective domains:
On the first piece:
which does fall in [-2, 2]. The FDT shows for less than and near -1, and for greater than and near -1, so is a local maximum.
On the second piece:
so it does not contribute any critical points.
Where the pieces meet:
By checking the conditions for continuity mentioned in part (a), we can determine that does not exist, but that doesn't rule out as a potential critical point.
We have
so for less than and near 1, and
so for greater than and near 1. So the FDT tells us that is a local minimum.
Finally, at the endpoints of the domain we're concerned with, [-2, 2]:
We have and .
So, on [-2, 2], attains an absolute minimum of and an absolute maximum of .