We characterize some major algorithmic randomness notions via differentiability of effective functions.
(1) As the main result we show that a real number
z
∈
[
0
,
1
]
z\in [0,1]
is computably random if and only if each nondecreasing computable function
[
0
,
1
]
→
R
[0,1]\rightarrow \mathbb {R}
is differentiable at
z
z
.
(2) We prove that a real number
z
∈
[
0
,
1
]
z\in [0,1]
is weakly 2-random if and only if each almost everywhere differentiable computable function
[
0
,
1
]
→
R
[0,1]\rightarrow \mathbb {R}
is differentiable at
z
z
.
(3) Recasting in classical language results dating from 1975 of the constructivist Demuth, we show that a real
z
z
is Martin-Löf random if and only if every computable function of bounded variation is differentiable at
z
z
, and similarly for absolutely continuous functions.
We also use our analytic methods to show that computable randomness of a real is base invariant and to derive other preservation results for randomness notions.