A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \in [a,b]$ such that $f(c)=\lambda$. Equivalently, the image of any interval under $f$ is an interval.

I know that there are functions $f: [0,1] \to [0,1]$ which have the Darboux property, but have no fixed points.

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> What are some articles which can be taken as references for this non-existence theorem? The only one I found was [this](https://projecteuclid.org/journals/real-analysis-exchange/volume-28/issue-2/An-example-of-a-Darboux-function-having-no-fixed-points/rae/1184963801.full), but I guess that there are older articles which deal with this subject. I searched Google and Mathscinet, but didn't find any except the one above (maybe I don't know how to search...).
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