- #1

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## 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?