Find points of discontinuity given g(x): x^2-1 for x <-1, 0 for -1< or equal to 1, x^2 for x>1. PLEASE explain steps.?

All parts of the piecewise definition are polynomials, hence continuous. If there is any discontinuity, it can occur only at an interval boundary. You can proceed from there after you have cleared up your function definition.

In the middle of the definition, you appear to be saying this:

g(x) = 0, for -1 ≤ 1

That is equivalent to saying that g(x) = 0 for all x on its domain.
That conflicts with the rest of the definition.

x^2 - 1 is smooth.
The constant 0 is smooth.
x^2 is smooth.

The only place it could be discontinuous is at the break points x = -1 or x = 1. See what the values of g(x) are on each side of those breaks and see if the curves match up.