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

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

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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 HeineĀBorel 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.]]>