Fixed point generalisation

Question

If a continuous function defined on an open interval in $R$ and the range is a proper subset of domain, then there exists a fixed point. Is this statement true? How about making domain and range compact and either of domain or range contains the other?

Answer 1

Try the function $f(x) = x/2$ on $(0,1)$.

It's true for a function mapping a compact interval into itself by the Brouwer fixed point theorem.