I know that the result is true for closed interval $[0,1]$ by using intermediate value property.
But in the case where we consider open interval $(0,1)$ does the solution change?
2026-04-02 16:51:20.1775148680
Does any continuous function from the open unit interval $(0,1)$ to itself has a fixed point?
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Try $x\mapsto x^2$ for a counterexample.