# 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?

Related Calculus and Beyond Homework Help News on Phys.org
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
Another approach: if f is continuous the inverse image of an open set is open, and the inverse image of a closed set is ....., and the subset {0} of the real numbers is ..... in the reals.

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?
Hold on a sec ...... Could you take a quick look at the solution on this UCLA Math website? http://www.math.ucla.edu/~elewis/Math230PDFs/Math%20230b%20HW1.pdf It's only 3 lines long and seems sketchy. How do we know that {0} is closed in our space X? If X = ℝ, then it would work out; but we're not given that.

Dick