Math Is Fun Forum

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

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 (2009-03-25 16:56:17)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Prove fixed point

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

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

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Prove fixed point

Apart from the fact that you had

amd

the wrong way round, you are correct.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

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.

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