a multiple choice question on monotone non-decreasing real-valued function on $\mathbb{R}$ let $f$ be a monotone non-decreasing real-valued function on $\mathbb{R}$ . Then
$1$. $\lim  _ {x \to a}f(x)$ exists at each point $a$.
$2$. If $a<b$ , then $\lim  _ {x \to a+}f(x) \le \lim  _ {x \to b-}f(x)$.
$3$. $f$ is an unbounded function.
$4$. The function $g(x)=e^{-f(x)}$ is a bounded function.

$2$ is looking obviously true.
but I need some counter example to disprove the others.can anyone help me please.thanks for your help.

1: monotone does not necessarily mean it 's continuous
3: never said it was strictly monotone, could be constant
4: $g$ is strictly decreasing and has a lower bound never reached ($0$). It has an upper bound only if $f$ has a lower bound -> many counter examples
