# Thread: Proof with closed sets....

1. ## Proof with closed sets....

Suppose f:[a,b]--> R and g:[a,b]-->R. Let T={x:f(x)=g(x)}
Prove that T is closed.

So we can set h(x)=f(x)-g(x) and then T is the set such that h(x)=0 and so T complement is the set with h(x)=/=0 and then if we can show that this is open, then it's complement, T is closed. I don't know how to show it is open though. We need to show there is some neighborhood, but I don't know what to do. Thanks.

2. Hello,

If f and g are continuous, then the proof is trivial, since for any continuous function h, $h^{-1}(\text{a closed sed})$ is a closed set.

And here, you have $h^{-1}(\{0\})$...