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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!

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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
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$x$ takes on values between $\alpha$ and $\beta$ –  user137794 May 11 at 22:25
    
Oops, thanks. My mistake –  Christopher Liu May 11 at 22:26

1 Answer 1

up vote 3 down vote accepted

The Conway base 13 function is one such function. From Wikipedia:

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

The construction of the function is a little bit complicated. Refer to the wiki page for details.

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I am really grateful for your prompt help! –  Clin May 11 at 22:41
    
@Clin Happy to help. –  Ayman Hourieh May 11 at 22:41

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