Let $a<b$ be real numbers, and let $f:[a, b] \rightarrow \mathbb{R}$ be continuous. Show that $f$ is bounded on $[a, b]$, and that there exist $c, d \in[a, b]$ such that for all $x \in[a, b]$, $f(c) \leqslant f(x) \leqslant f(d)$.

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that

$$\lim _{x \rightarrow+\infty} g(x)=\lim _{x \rightarrow-\infty} g(x)=0$$

Show that $g$ is bounded. Show also that, if $a$ and $c$ are real numbers with $0<c \leqslant g(a)$, then there exists $x \in \mathbb{R}$ with $g(x)=c$.