Real Analysis - Combinations of Continuous Functions #2

Quote:

Originally Posted by ajj86

Assume h: R --> R is continuous on R and let K = {x: h(x) = 0}. Show that K is a closed set.

It is a closed set because it's the reciprocal image of a closed set by a cont. application.

Proof :
Let's show that the reciprocal image by a continuous application of an open set is open.
Let $x \in f^{-1}(O)=\{x \in R ~:~ f(x) \in O\}$ , where O is an open set. Because it's open, it's a neighbourhood of any of its point. Hence it is a neighbourhood of f(x).
Because the function is continuous, $f^{-1}(O)$ will be a neighbourhood of x.
Hence $\forall x \in f^{-1}(O)$ , $f^{-1}(O)$ is a neighbourhood of x. So it's a neighbourhood of all its points. So it's an open set.

Now use Hausdorff's formula : $f^{-1}(^cO)=^c f^{-1}(O)$
$^cO$ is a closed set.
$f^{-1}(O)$ is open, and therefore $^c f^{-1}(O)$ is a closed set.

We're done.

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$K=h^{-1}(\{0\})$
and in the usual topology over R, {0} is a closed set.

Quote:

Aside: (In the past we have used K to denote a compact set. I'm not sure if it is denoting one here or not. Would they have to specify in the beginning of the problem that K is compact?)

No, but they could ask you to show it is a compact.