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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.