The Cantor staircase function 
<https://en.wikipedia.org/wiki/Cantor_function>
has an interesting property: $\{x\colon f'(x)\neq 0\}$ is a nowheredense nullset.
But it it differentiable almost everywhere, not everywhere.
Hence my question is if nondecreasing nonconstant differentiable function can have the above property?