Final answer:
The function g(x, y) = ln (x² y² - 4) is continuous for all points (x, y) where y > 2/|x| or y < -2/|x|, and x is not zero, as these conditions keep the argument of the natural logarithm positive.
Step-by-step explanation:
To determine the set of points at which the function g(x, y) = ln (x² y² - 4) is continuous, we need to identify the domain of the function first. The natural logarithm function ln(z) is defined for all z > 0. Thus, the inside of the logarithm, x² y² - 4, must be greater than zero. This gives us the inequality:
x² y² > 4.
Solving for y, we have:
y² > 4/x²
Taking the square root of both sides, we find that y > 2/|x| or y < -2/|x|. This means that the function g(x, y) will be continuous for all points (x, y) where y is exclusively within these bounds and x is not zero. To put it another way, the continuity of g(x, y) is determined by the set of points that lie outside of the region enclosed by the hyperbolas y = ±2/|x|. Additionally, x cannot be zero, as this would make the denominator of the expression undefined.
Therefore, the function is continuous on the set of points in the xy-plane that satisfy y > 2/|x| or y < -2/|x| where x ≠ 0.