Determine the set of points at which the function is continuous

Question

G(x, y) = ln(x² + y² − 4)

Answer 1

Answer 2

Any time we're talking about discontinuities, we need to go back to Algebra 2 and look at restrictions on the domain. This being a multivariate function doesn't change that. If you can, picture this as a 3D surface where you're looking over an x,y plane. Any value of (x, y) that doesn't exist is a discontinuity.

In this case, ln(x) has the restriction that its argument must be positive. Therefore:

x² + y² - 4 > 0

Add 4 to both sides and we get what looks like a circle equation with a radius of 2.

x² + y² > 2²

The stupid me can't figure out these inequalities, so I just plug in a point inside the circle and if it doesn't work, I include all the points outside of it:

1² + 1² > 4 [false]

Therefore everything outside the circle is fine.

If you try a few of these points on the 3D picture, you should get a black-hole-looking thing with an infinite drop right around where the circle is