Blancmange curve ($T(x)={\sum }_{j\ge 0}{2}^{-j}\mathrm{dist}(2^{j}x,\mathbb{Z})$)

The Takagi function is a continuous nowhere differentiable function. The triangular summands are continuous and the geometric tail is bounded uniformly. On dyadic intervals of length $2^{-N}$, all terms of index at least $N$ vanish at both endpoints, while every lower summand has slope $+1$ or $-1$. Each refinement therefore changes the secant slope by exactly one in absolute value. The secants straddling a differentiability point lemma rules out a derivative at every real point.