Question

At what points (x,y,z) in space are the following functions continuous?

a. h₁(x,y,z)=xy+1/z+6
b. h₂(x,y,z)=1/x²+z²-5

Answer 1

Function h₁(x,y,z) is continuous for all values of x and y, but z cannot be equal to 0. Function h₂(x,y,z) is continuous for all values of x, y, and z, except x cannot be equal to 0.

Step-by-step explanation:

For function h₁(x,y,z)=xy+1/z+6 to be continuous, each term in the function must be continuous.

The product of x and y is continuous for any value of x and y, as the product of two continuous functions is also continuous.

The term 1/z is continuous for all values of z except z=0. So, z cannot be equal to 0 for the function to be continuous.

The constant term 6 is always continuous.

Therefore, function h₁(x,y,z) is continuous for all values of x and y, but z cannot be equal to 0.

For function h₂(x,y,z)=1/x²+z²-5, each term is continuous.

The term 1/x² is continuous for all values of x except x=0.

The term z² is continuous for all values of z.

And the constant term -5 is always continuous.

Therefore, function h₂(x,y,z) is continuous for all values of x, y, and z, except x cannot be equal to 0.