The Denjoy alternative for computable functions

Overview & Synopsis

The Denjoy-Young-Saks Theorem from classical analysis states that for an arbitrary real-valued function f, the Denjoy alternative holds outside a null set. This means that for almost every real x, either the derivative of f exists at x, or the derivative fails to exist in the worst possible way: the slopes of f around x take arbitrarily large positive values and arbitrarily large negative values. Demuth studied effective versions of this theorem, in particular the effective version when the function f is Markov computable. He then looked at the set DA of reals x such that any Markov computable function satisfies the Denjoy alternative at x. He introduced a notion of algorithmic randomness (now known as Demuth randomness) which he proved to be sufficient for a point to belong to DA. In this paper, we show that DA is in fact strictly contained in the set of Demuth random points. We also obtain interesting side results about effective versions of Lebesgue's density theorem.