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?
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 $x$.
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".
