I was taught that you don't talk about a discontinuity in points were the function is not defined. As in this example the function is not defined in x = 0. Hence we don't have a discontinuity here.

But at x = 1 we have a discontinuity. The book told us it was a removable discontinuity, and in this point we don't have lim{x->1} = f(1).

For the second question we need to know that

lim{x->a} f(x) = L <=> lim{x->a+} f(x) = lim{x->a-} f(x) = L

to consider the function having a limit L at x = a.

Thus we have no limit att x = 0 and neither at x = 1.