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## #1 2009-03-11 22:25:17

JaneFairfax
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### Prove fixed point

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 Heine–Borel 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 (2009-03-26 15:56:17)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #2 2009-03-12 01:40:17

mathsyperson
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### Re: Prove fixed point

Last edited by mathsyperson (2009-03-12 03:24:31)

Why did the vector cross the road?
It wanted to be normal.

## #3 2009-03-12 03:14:21

JaneFairfax
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### Re: Prove fixed point

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

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #4 2009-03-12 03:23:24

mathsyperson
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### Re: Prove fixed point

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.