  We consider effective versions of two classical theorems, the Lebesgue
density theorem and the Denjoy-Young-Saks theorem. For the first, we show that
a Martin-Loef random real $z\in [0,1]$ is Turing incomplete if and only if
every effectively closed class $C \subseteq [0,1]$ containing $z$ has positive
density at $z$. Under the stronger assumption that $z$ is not LR-hard, we show
that $z$ has density-one in every such class. These results have since been
applied to solve two open problems on the interaction between the Turing
degrees of Martin-Loef random reals and $K$-trivial sets: the non-cupping and
covering problems.
  We say that $f\colon[0,1]\to\mathbb{R}$ satisfies the Denjoy alternative at
$z \in [0,1]$ if either the derivative $f'(z)$ exists, or the upper and lower
derivatives at $z$ are $+\infty$ and $-\infty$, respectively. The
Denjoy-Young-Saks theorem states that every function
$f\colon[0,1]\to\mathbb{R}$ satisfies the Denjoy alternative at almost every
$z\in[0,1]$. We answer a question posed by Kucera in 2004 by showing that a
real $z$ is computably random if and only if every computable function $f$
satisfies the Denjoy alternative at $z$.
  For Markov computable functions, which are only defined on computable reals,
we can formulate the Denjoy alternative using pseudo-derivatives. Call a real
$z$ DA-random if every Markov computable function satisfies the Denjoy
alternative at $z$. We considerably strengthen a result of Demuth (Comment.
Math. Univ. Carolin., 24(3):391--406, 1983) by showing that every Turing
incomplete Martin-Loef random real is DA-random. The proof involves the notion
of non-porosity, a variant of density, which is the bridge between the two
themes of this paper. We finish by showing that DA-randomness is incomparable
with Martin-Loef randomness.
