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Bits: 4.Q:
Preimage of a compact set has a open set
Let $K \subset \mathbb{R}$ be a compact and $f : \mathbb{R} \to \mathbb{R}$ a continuous function. Show that there exists a $g: K \to \mathbb{R}$ such that the preimage of a compact set $K$ has an open set.
My work: For the set $g(K)$ to be open we need to find an open set $O \supset g(K)$. It is also assumed that $K$ is closed (closed in $\mathbb{R}$ and not $\mathbb{R}$) and therefore $K \subset \mathbb{R}$. By compactness, there exists a finite set of elements $X=\{x_1, x_2,…, x_n\}$ s.t $K \subset \cup_{i=1}^n B(x_i, \epsilon)$ for a very small $\epsilon$. I’m not sure what to do from here though. I was thinking that I could say something like $B(x, \epsilon) \supset \cup_{i=1}^n B(x_i, \epsilon)$, but that doesn’t really give me much insight.
Thanks for your help.
A:
Consider the functions $\phi_n(x)=\frac 1 n \mathbf{1}_{(x-n^{ -1},x)}$.
Then $f(x)=\lim \phi_n(x)$ and $\phi_n$ are continuous and their limit is continuous, hence $f$ is continuous.
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