At what points (x, y, z) in space is the function f(x,y,z)=x^ 2 + y ^2 - 2z^2 continuous?

Question

how to find this?????what is the procedure of this questiion??????

Answer 1

There are three rules a function must abide by if it is to be considered continuous at a point:

1. It must be defined at that point

2. The limit must exist at the point

3. The limit at the point must equal the function evaluated at that point

 

For example, the point (0,0,0) for this function:

1. There are no rational expressions that would make (0,0,0) be undefined

2. The limit exists

3. The limit at (0,0,0) = 0, and f(0,0,0) = 0

Therefore, the function is continuous at (0,0,0)

 

In looking at the equation, there aren't any rational expressions or roots that would cause any discontinuity, so the function should be continuous at all points on the xyz-plane.