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Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous bounded function, then: 

  1. $f$ has to be uniformly continuous 
  2. There exists an $x \in \mathbb{R}$ such that $f(x) = x$ 
  3. $f$ cannot be increasing 
  4. $\lim_{x \rightarrow \infty} f(x)$ exists.
asked in Set Theory & Algebra by Veteran (29.2k points)   | 32 views

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