integration - anti-derivative not differentiable at any point

Reading about primitives and anti-derivatives, I noticed that primitive functions of non-continuous functions are not differentiable at some point, but the set of non-differentiability is often negligible.

I tried to think of a function horrible enough to get a non-differentiable antiderivative, and I found some with non-negligible set of non-differentiability points.

But I never found an antiderivative that is nowhere differentiable. Can you find one?

Answer

It isn't possible. Lebesgue's Differentiation Theorem states that if $f$ is integrable over $\mathbb{R}$, and we let:

$$ F(x) = \int \limits_{(-\infty, x]} f(t) \, dt $$

Then $F$ is almost everywhere differentiable.