# Math Help - range of unbounded function

1. ## range of unbounded function

The function $h:\mathbb{R}\rightarrow\mathbb{R}$ is continuous on $\mathbb{R}$ and let
$h\left (\mathbb{R} \right )= \left\{h(x):x\in\mathbb{R} \right \}$
be the range of $h$. Prove that if $h\left (\mathbb{R} \right )$ is not bounded above and not bounded below, then $h\left (\mathbb{R} \right )=\mathbb{R}$.

I tried using the definitions of not bounded above and not bounded below, i.e.

not bounded above: $\exists{u}\in\mathbb{R}, \forall{s}\in\mathbb{R}\Rightarrow{s}>{u}$
not bounded below: $\exists{v}\in\mathbb{R}, \forall{v}\in\mathbb{R}\Rightarrow{s}<{v}$

Am I on the right track? How do I proceed from here?

2. ## Re: range of unbounded function

If $y\in\mathbb{R}$ , as $h(\mathbb{R})$ is not bounded then there exist $x_1,x_2\in\mathbb{R}$ such that $h(x_1) . Now, apply the Intermediate Value Theorem .

3. ## Re: range of unbounded function

Ok. I got it thanks!