How to extend Solomonoff Induction to continuous domain
The set of computable real functions is also recursively enumerable. Assume it is enumerated by $(f_i)_{i\in\omega}$. Then the Solomonoff prior of $f_i$ is $p(f_i)=2^{-K(i)}$. So there is no difference whether the function is defined over natural numbers or real numbers, as long as it is computable.