Final answer:
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.