The distribution of binary digits of integers.

In 1968, Trollope proved the following result: Let . A typical integer has digits, half of which one expects to be 1's, so that should be approximately . Therefore, we compare with , when and have the same fractional part, by considering the function

for each . One can easily show that this limit exists and that the function is continuous. However Trollope proved the surprising result that is nowhere differentiable. For more on such questions see the paper by Boyd, Cook and Morton.