Function which graph looks looks like the plane 
Does there exist a function which is surjective from $[a,b]\to\mathbb R$ for any $a,b\in\mathbb R$ such that $a\ne b$?

Of course such a function would have a graph which looks like the plane.
I can see that it is, of course, not continuous.
Do you have any examples of such functions? Do they exist?
 A: Yes, if we're allowed to rely on the axiom of choice.
Let $\sim$ be the equivalence relation on $\mathbb{R}$ given by $x\sim y$ iff $x-y\in\mathbb{Q}$. Since each equivalence class is countable, the set of equivalence classes (a.k.a. the quotient $\mathbb{R}/{\sim}$) has the same cardinality as $\mathbb{R}$, hence there is a bijection $g\colon\mathbb{R}/{\sim}\to\mathbb{R}$. Now let $f(x)=g([x])$, where $[x]$ is the equivalence class of $x$.
A: If you assume the Axiom of Choice - the non-linear solutions to Cauchy's functional equation may be an interesting candidate. 
Some of the interesting properties which such solutions possess are :


*

*They are discontinuous everywhere (nowhere continuous)

*They are unbounded on any segment, no matter how small (nowhere locally bounded)

*From the above it also follows that it is nowhere locally monotone.

*They are dense in $\mathbb{R^2}$
You can read up on Cauchy's Functional Equation on Wikipedia :
http://en.wikipedia.org/wiki/Cauchy%27s_functional_equation
Here's an article from a blogger on the non-linear solutions to CFE :
http://www.cofault.com/2010/01/hunt-for-addictive-monster.html 
The above article also gives a nice construction of such a "monster".
