= Blancmange curve
{title2=$T(x)=\sum_{j\ge0}2^{-j}\operatorname{dist}(2^jx,\mathbb Z)$}
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= Takagi function
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{synonym}
= Blancmange function
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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.
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