Random Continuous Functions

We investigate notions of algorithmic randomness in the space C(N2) of continuous functions on N2. A probability measure is given and a version of the Martin-Löf test for randomness is defined which allows us to define a class of (Martin-Löf) random continuous functions. We show that random continuous functions exist, but no computable function can be random. We show that a random function maps any computable real to a random real and that the image of a random continuous function is always a perfect set and hence uncountable. We show that for any y∈N2, there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set.