calculus lll?

Since the numerator and denominator of f are polynomials, it follows that f is continuous whenever its denominator is not equal to 0:
y^2 - x^2 - 1 ≠ 0
==> y^2 - x^2 ≠ 1.

This is all of R^2, except for the hyperbola y^2 - x^2 = 1 (draw with a dashed line), which has its vertices at (0, 1) and (0, -1) and has asymptotes y = x and y = -x.

I hope this helps!