Brattka, Miller and Nies proved in 2011 that a real z is computably random iff every nondecreasing function is differentiable at z. I will consider the analogous theorem when the effectiveness condition on the function is varied.
1. The analogous theorem holds for polynomial time randomness and polynomial time computable nondecreasing functions.
2. For interval-c.e. function (essentially, the variation function of a computable function), the right randomness strength is Martin-Loef random reals at which the Lebesgue density theorem holds for effectively closed sets.
Surprisingly, the analytic notion of porosity plays a major role in both proofs.