Continuity with multible vars?

Question

Determine the set of points at which the function f(x; y; z) = square root(x + y + z) is continuous.

How are you supposed to determine the interval of continuity like that? I can pick random numbers and try them out but it really doesn't help, it will just take forever.

Answer 1

First of all, the function is only well-defined in the region where x + y + z is non-negative. So the region of continuity is contained in that. In fact, it's contained in the interior of that region -- i.e., excluding the boundary set, namely the points where x + y + z = 0.

Second, all three partial derivatives are well-defined in the region where x + y + z is positive. So the region of continuity includes that.

But I've just said that the region of continuity both contains and is contained in the set of points where x + y + z = 0. I.e., it equals that set.

So we're done!

So you have your answer