Let f be a continuous real function on a metric space X. Let

Homework Statement

Let f be a continuous real function on a metric space X. Let Z(f) be the set of all p in X at which f(p) = 0. Prove that Z(f) is closed.

Homework Equations

Definition of continuity on a metric space.

The Attempt at a Solution

Proof. We'll show that X/Z(f) = {p in X s.t. f(p) ≠ 0} is open. Choose p in X/Z(f). Since f is continuous, for every ε > 0 there exists a ∂ > 0 such that d(f(x),f(p)) < ε whenever d(x, p) < ∂.

..... Unfortunately, I suffered a brain hemorrhage before I could finish this. I think I was trying to show that p is an interior point of X/Z(f). Thoughts?

Dick
Homework Helper
I'd look into getting the brain hemorrhage treated. What might be confusing you is that you are using 'd' for the metric on R as well as on X. How about writing the metric on R as |f(x)-f(p)|<epsilon. Now pick epsilon=|f(p)|/2. Can f(x) be zero?

Bacle2