# construct a function

I wonder whether there is a function $f\colon\Bbb R\to\Bbb R$ with the folowing characteristic? for every two real numbers $\alpha,\beta,\alpha\lt\beta$, $$\{f(x):x\in(\alpha,\beta)\}=\Bbb R$$

I can't say such a function does not exist, neither can I construct a example

Thanks a lot!

-
How is this a function from $\mathbb{R} \to \mathbb{R}$? $x$ seems to take on values in $\mathbb{R^2}$. –  Christopher Liu May 11 at 22:24
$x$ takes on values between $\alpha$ and $\beta$ –  user137794 May 11 at 22:25
Oops, thanks. My mistake –  Christopher Liu May 11 at 22:26

$f$ takes as its value every real number somewhere within every open interval $(a,b)$.