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)