Exercise 14, § 6

Let $f$ be a continuous mapping of $\mathbf{R}$ into itself. a) Show that if $f$ is uniformly continuous on $\mathbf{R}$, then there exist two real numbers $\alpha \geq 0$ and $\beta \geq 0$ such that $|f(x)| \leq \alpha|x| + \beta$ for all $x \in \mathbf{R}$. b) Show that if $f$ is monotone and bounded in $\mathbf{R}$, then $f$ is uniformly continuous on $\mathbf{R}$.