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Let $f:\mathbb R \to \mathbb R$ be a differentiable function . If $\exists r \in \mathbb R $ such that $|f'(x)|\le r<1 , \forall x \in \mathbb R$ then using Lagrange's theorem one can show $f$ is a Lipscitz contraction and then use Banach contraction principle to conclude $f$ has a unique fixed-point. My question is what happens if $f'(x)\le r<1 , \forall x \in \mathbb R$ ? Then does $f$ even have a fixed point ?