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Please help me do the following.

Suppose that $f:\mathbb{R}^m\to\mathbb{R}$ satisfies two conditions:

(i) For each compact set $K$, $f(K)$ is compact.

(ii) For any nested decreasing sequence of compacts $(K_n)$, $$f\left(\bigcap K_n\right)=\bigcap f(K_n).$$ Prove that $f$ is continuous.

Property (ii) implies the following: If $(x_i)$ is a sequence in $\mathbb{R}^m$ converging to $x$ such that $f(x_1)=f(x_2)=\ldots$, then $f(x)=f(x_1)$. I think they are in fact equivalent, but I'm not sure.

Edit: Sorry for the confusion, the second condition is now corrected.

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    Excuse me, english is not my native language. What is a nested decreasing sequence of compact sets?2012-04-23
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    Are you sure this is a union in (ii)?2012-04-23
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    @matgaio Each $K_n$ is compact and $K_{n+1}\subset K_n$. On another note, the union in (ii) should probably be an intersection. Indeed,if one of $K_n=\varnothing$, then the LHS is empty, whereas the RHS is nonempty if at least one of the $K_n$ is nonempty.2012-04-23
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    I think you mean $f(\cap K_n) = \cap f(K_n)$ in ii.2012-04-23
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    I thought the same about the condition ii), in order to get some convergence property to this function.2012-04-23
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    Indeed, I made a typo.2012-04-23

2 Answers 2