Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous bounded function, then:
- $f$ has to be uniformly continuous
- There exists an $x \in \mathbb{R}$ such that $f(x) = x$
- $f$ cannot be increasing
- $\displaystyle \lim_{x \rightarrow \infty} f(x)$ exists
