(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Letfbe a continuous real function on a metric spaceX. Let Z(f) be the set of allpinXat whichf(p) = 0. Prove that Z(f) is closed.

2. Relevant equations

Definition of continuity on a metric space.

3. The attempt at a solution

Proof.We'll show thatX/Z(f) = {pinXs.t.f(p) ≠ 0} is open. ChoosepinX/Z(f). Sincefis continuous, for every ε > 0 there exists a ∂ > 0 such thatd(f(x),f(p)) < ε wheneverd(x,p) < ∂.

..... Unfortunately, I suffered a brain hemorrhage before I could finish this. I think I was trying to show thatpis an interior point ofX/Z(f). Thoughts?

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

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