chapter  7
Continuous nowhere approximately differentiable functions
ByGilbert W. Bassett Jr., Roger Koenker
Pages 12

The first example of a nowhere approximately differentiable function belonging to the Banach space C[0, 1] is due to Jarník (see [95]). Moreover, he showed that such functions are typical, i.e., they constitute a set whose complement is of first category in C[0, 1].