I believe the [Takakgi function](https://en.wikipedia.org/wiki/Blancmange_curve) satisfies this property. According to [The Takagi Function: A Survey](https://arxiv.org/abs/1110.1691v2), the Takagi function $T$ satisfies
$$T(x+h)-T(x) = O(h\log(1/|h|) \: \text{ as } \: h\to0$$
and this is the best possible estimate.