computability theory - Differentiability of computable functions

You may be interested in some very recent work by Brattka, Miller and Nies looking at points of differentiability for computable functions in terms of algorithmic randomness. Briefly call a real x computably random (Martin-Löf random) if no computable (computably enumerable) martingale succeeds on a binary representation of x. Brattka, Miller and Nies show that:

1) At each computably random real, every computable function that is non-decreasing is differentiable.

2) At each Martin-Löf random real, every computable function of bounded variation is differentiable.