Fixed point theorem on disjoint interval

Question

I am to prove that there exists $x^*\in [0,1] \cup [2,3]$ for a function $f$ from $[0,1] \cup [2,3] \to [0,1] \cup [2,3]$ s.t $f(x^*) = x*$ basically the fixed point theorem for the above disjoint interval. It is given that $f$ is continous.

My attempt : I tried going the standard route and define $g(x) = f(x) - x$ then i say that $g(0) > 0$ for $g(1)$ we have two cases it can be either positive or negative, if it is negative then proof is done by IVT, if it is positive then I check $g(2)$ which again has the same two cases, now if $g(2)$ is also positive then I check for $g(3)$ and I say that $g(3)$ has to be negative. So now I have $g(0) > 0$ and $g(3) < 0$ so by invoking IVT I could say that there exists a point s.t. $f(x^*) = x^*$ but since the interval is disjoint I am not sure that I can use IVT here( or maybe the function being continuous allows me to???).

So can someone help me out in this and let me know if there is a proof for the above problem if my proof is incorrect.

The fixed point theorem doesn't hold on sets such as these. Start looking for a counterexample. — Theo Bendit, May 28 '19 at 06:34

score 1 · Accepted Answer · answered May 28 '19 at 06:35

1

The claim in the question is not true in general: let $f(x)=3-x$ for $x$ in its domain, then an $x^*$ does not exist.

answered May 28 '19 at 06:35

Parcly Taxel

105,904

Ok thanks, so it can be both true and false depending on the scenario – user601297 May 28 '19 at 06:38
@user601297 Or rather, sometimes true sometimes false. – Parcly Taxel May 28 '19 at 06:38
what can we say about the fixed point theorem on $(0,1)$ where $f$ is continuous and what about on [0,1] but $f$ is discontinuous??? – user601297 May 28 '19 at 06:46
@user601297 Please ask that in another question, and upvote and accept my answer. – Parcly Taxel May 28 '19 at 06:46
@user601297 https://math.stackexchange.com/questions/3114789/let-fx-to-x-be-continuous-does-f-have-a-fixed-point-when-x-0-1-or-x/3114810#3114810 – Theo Bendit May 28 '19 at 07:49

Fixed point theorem on disjoint interval

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