$f: [a, b] \mapsto [a,b]$, $f$ is monotonically increasing $\Rightarrow \exists~ x \in (a, b)$ such that $x = f(x)$

Question

I can prove this, if $f$ is continuous on $[a, b]$, using $g(x) = f(x) - x$. But can't figure out, how can I prove it, if $f(x)$ is only monotonically increasing.

Answer 1

Consider $X = \{x ∈ [a..b];~f(x) ≥ x\}$. Now $X ≠ ∅$, as $a ∈ X$, so you may consider $ξ = \sup X$.

• Generally, $f(X) ⊆ X$, as for $x ∈ X$, by monoticity $f(x) ≥ x \implies f(f(x)) ≥ f(x)$.
• You have $f(ξ) ≥ f(x) ≥ x$ for all $x ∈ X$ by monoticity of $f$, so $f(ξ) ≥ \sup X = ξ$.
• But then $ξ ∈ X$, so …