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Let $f: \mathbb{R} \to \mathbb{R}$. Prove that if $f$ is bounded such that for all $x, y \in \mathbb{R}$ $x\neq y$ implies $|f(x) - f(y)| \lt |x -y|$ and for all $x \in \mathbb{R}$, $f$ is differentiable at $x$ with $|f'(x)| \lt 1$ then $f(z) = z$ for exactly one number $z \in \mathbb{R}$.

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    mary = Omar? http://mathoverflow.net/questions/600542011-03-30
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    Please don't ask question in imperative mode. Also, it would be nice if you included your work on the problem in question.2011-03-30

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