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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

This is an adaptation of a problem I came across on another forum. The problem was originally stated more generally as follows: if

is a compact subset of with the usual topology and is continuous with , then has a fixed point. Unfortunately this is not quite true: needs to be connected as well as compact. Since, by the HeineBorel theorem, any compact and connected subset of is a closed and bounded interval, I have chosen the interval for convenience.Note also a slight difference in this problem from the originally stated one. In the original problem, the domain of

is a subset of the range of . In my adaptation of the problem, the reverse is the case.*Last edited by JaneFairfax (2009-03-25 16:56:17)*

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

*Last edited by mathsyperson (2009-03-11 04:24:31)*

Why did the vector cross the road?

It wanted to be normal.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Apart from the fact that you had amd the wrong way round, you are correct.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Ah, so I have. Fixed now, thanks.

I wrote g(x) differently to how I meant it, which is what caused those intervals to be the wrong way round.

Why did the vector cross the road?

It wanted to be normal.

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