We investigate the notion of K-triviality for closed sets and continuous functions in the Cantor space . For every K-trivial degree, there exists a closed set and a continuous function of the same degree. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial effectively closed set with no computable elements. A closed set is K-trivial if and only if it is the set of zeroes of some K-trivial continuous function. We also give a density result for the Medvedev degrees of K-trivial effectively closed sets. Finally, we give a completeness criterion related to randomness of closed sets.