Questions about $f: \mathbb{R} \rightarrow \mathbb{R}$ with bounded derivative I came across a problem that says:
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function. If $|f'|$ is bounded, then which of the following option(s) is/are true?

(a) The function $f$ is bounded.
  (b) The limit $\lim_{x\to\infty}f(x)$ exists.
  (c) The function $f$ is uniformly continuous.
  (d) The set $\{x \mid f(x)=0\}$ is compact.

I am stuck on this problem. Please help. Thanks in advance for your time.

(a) & (b) are false: Consider $f(x)=x$ $\forall$ $x\in\mathbb R$;
(c) is true: $|f'|$ is bounded on $\mathbb R\implies f'$ is bounded on $\mathbb R$ [See: Related result];
(d) is false: $f=0$ on $\mathbb R\implies${$x:f(x)=0$} $=\mathbb R$.
