Randomly generate all functions from R to R

Say I want a random generator of functions like this:

f: R -> R
   x -> f(x)

It can be usual functions of course, like sin(x) or exp(x) for instance. But what about functions that are not defined and can not be defined from usual functions. Where would you start to build such a generator?

Important precision: here I am asking about all functions, but for my application I'm really only interested in continuous strictly monotonic functions. So if you have an answer for that, I'll already be satisfied.