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Let $f \colon \mathbb{R}→ \mathbb{R}$ be a continuous function.Which of the following is always true?
1. $f ^{-1}(U)$ is open for all open sets $U ⊆\mathbb{R}$
2. $f ^{-1}(C)$ is closed for all closed sets $C ⊆\mathbb{R}$
3. $f ^{-1}(K)$ is compact for all compact sets $K⊆ \mathbb{R}$
4. $f ^{-1}(G)$ is connected for all connected sets $G ⊆ \mathbb{R}$

1 and 2 are always true but i am not sure about the others.can somebody help me

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Have you tried looking at different examples of continuous functions to see if 3 and 4 are true for those examples? –  Jonas Meyer Dec 21 '12 at 2:17

2 Answers 2

up vote 9 down vote accepted

(3) is not true. Consider the function $f(x)=0$ for $x\in\mathbb R$. $\{0\}$ is compact but $f^{-1}(0)=\mathbb R$ is not.

(4) is not true. Consider the function $f(x)=(x-2)(x+2)$ for $x\in\mathbb R$. $\{0\}$ is connected but $f^{-1}(0)=\{-2,2\}$ is not.

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I would have used the sine function for my counterexample for 3 and 4. –  Lubin Apr 7 '13 at 2:34

(3) What if $f(x)=0$ for all $x\in\Bbb R$?

(4) What if $f(x)=x^2$ and $G=\{1\}$?

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