# Thread: range of unbounded function

1. ## range of unbounded function

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

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

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

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

2. ## Re: range of unbounded function

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

3. ## Re: range of unbounded function

Ok. I got it thanks!