Algorithmic randomness of continuous functions

We investigate notions of randomness in the space C(2N) of continuous functions on 2N. A probability measure is given and a version of the Martin-Löf Test for randomness is defined. Random ∆2 continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any y ∈ 2N, 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. The set of zeroes of a random continuous function is always a random closed set.