Showing a set is compact

Homework Statement

Explain why $f(x,y,z) = x + y - z$ must attain both a maximum and a minimum on the sphere $x^2 + y^2 + z^2 = 81)$.

None

The Attempt at a Solution

I know that any continuous function attains both a maximum and a minimum on a compact set. I defined the set $K$ as $$K = \{(x,y,z)|g(x,y,z) = 0\}$$ and demonstrated that it is bounded by a closed ball. I then realized, however, that I'm not really sure how to show that the set is closed. I understand the definition of a closed set, but I don't know how to show that it is true in this particular case. Any hints?

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pasmith
Homework Helper

Homework Statement

Explain why $f(x,y,z) = x + y - z$ must attain both a maximum and a minimum on the sphere $x^2 + y^2 + z^2 = 81)$.

None

The Attempt at a Solution

I know that any continuous function attains both a maximum and a minimum on a compact set. I defined the set $K$ as $$K = \{(x,y,z)|g(x,y,z) = 0\}$$
What is g?

and demonstrated that it is bounded by a closed ball. I then realized, however, that I'm not really sure how to show that the set is closed. I understand the definition of a closed set, but I don't know how to show that it is true in this particular case. Any hints?
It suffices to show that the sphere is compact. To do that, you can write down a continuous surjection from a compact subset of the plane to the sphere and use the result that the continuous image of a compact space is compact.

Mark44
Mentor

Homework Statement

Explain why $f(x,y,z) = x + y - z$ must attain both a maximum and a minimum on the sphere $x^2 + y^2 + z^2 = 81)$.

None

The Attempt at a Solution

I know that any continuous function attains both a maximum and a minimum on a compact set. I defined the set $K$ as $$K = \{(x,y,z)|g(x,y,z) = 0\}$$ and demonstrated that it is bounded by a closed ball. I then realized, however, that I'm not really sure how to show that the set is closed. I understand the definition of a closed set, but I don't know how to show that it is true in this particular case. Any hints?
What's g(x, y, z)?

Oops, I'm sorry, I defined $g$ as $g(x,y,z) = x^2 + y^z + z^2 - 81$. So the constraint equation.

We haven't demonstrated that compactness is continuous invariant in class, unfortunately, so I would be required to prove that. If demonstrating the set is closed is more complicated, I'll do it, but I would rather not prove a theorem if it's not strictly necessary.

Dick
Oops, I'm sorry, I defined $g$ as $g(x,y,z) = x^2 + y^z + z^2 - 81$. So the constraint equation.