Characterize those functions which maps open interval to open interval(s).

Question

This question was asked by one of my Professor during the class. I'm getting intuition that these functions should be one-one (I'm wrong maybe). But, I'm unable to classify all such functions.

Please help in this!!

Answer 1

HINT: Not all bijections work, but you should be able to prove that the monotone (increasing or decreasing) bijections do. You should also be able to prove that these are precisely the continuous bijections. Going a bit further, try to prove that if $f:\Bbb R\to\Bbb R$ has an absolute maximum or minimum on some open interval $(a,b)$, then $f$ does not have the desired property, and use this to show that the continuous bijections are the only continuous functions that work.

However, you won’t be able to characterize these functions completely, because (assuming the axiom of choice) there are wildly discontinuous functions, impossible to visualize, that take open intervals to open intervals — that in fact map every open interval onto a single open interval. The construction of such a function requires a bit of set-theoretic background that you quite likely don’t have, but I’m going to include it for the sake of completeness.

Let $\{\langle a_\xi,b_\xi,c_\xi\rangle:\xi<2^\omega\}$ be an enumeration of $\{\langle a,b,c\rangle\in\Bbb R^3:a<b\}$, and let $\{u_\xi:\xi<2^\omega\}$ be an enumeration of the open interval $(0,1)$. Suppose that $\eta<2^\omega$, and we’ve chosen distinct $x_\xi\in(a_\xi,b_\xi)$ for $\xi<\eta$. Let

$$\alpha=\min\big\{\beta<2^\omega:u_\beta\in(a_\eta,b_\eta)\setminus\{x_\xi:\xi<\eta\}\big\}\;,$$

and set $x_\eta=u_\alpha$; this is always possible, since $|\{x_\xi:\xi<\eta\}|=|\eta|<2^\omega=|(a_\eta,b_\eta)|$. Thus, the recursion goes through to $2^\omega$, and for each $\xi<2^\omega$ we have a distinct $x_\xi\in(a_\xi,b_\xi)$.

Now let $X=\{x_\xi:\xi<2^\omega\}$ and define

$$f:\Bbb R\to(0,1):x\mapsto\begin{cases} c_\xi,&\text{if }x=x_\xi\\ 1/2,&\text{if }x\in\Bbb R\setminus X\;. \end{cases}$$

Let $(a,b)$ be any open interval in $\Bbb R$ and $c$ any real number in $(0,1)$; then $\langle a,b,c\rangle=\langle a_\xi,b_\xi,c_\xi\rangle$ for some $\xi<2^\omega$, we chose $x_\xi$ so that $x_\xi\in(a_\xi,b_\xi)=(a,b)$, and we defined $f$ so that $f(x_\xi)=c_\xi=c$. Thus, $f[(a,b)]=(0,1)$: $f$ maps every open interval of $\Bbb R$ onto the interval $(0,1)$.

Note that we can replace $(0,1)$ in this argument with any other open interval of $\Bbb R$, including $\Bbb R$ itself.

Answer 2

What have you tried so far?

And, as for being one-to-one, can you think of examples where a function maps, say, every open interval to $R$? what would such a function look like? is it one-one? onto?