The longest path connecting (0, 0) and (1,1), with $f'(x)\ge0$?
Considering that you want a function that is differentiable throughout with $f'(x) > 0$, this rules out any arbitrarily defined zigzag paths or convolutions. One solution in this case would be $$f(x) = \lim_{n \to \infty} x^n, n \in N$$ This would be pretty close to the black path you have traced out, and would also satisfy the conditions, given that $$f'(x) = nx^{n-1} > 0 \ \ \forall \ x \in (0, 1)$$