# Approximation of the fractal Takagi (blancmange) curve

The sum of very simple "sawtooth functions" (see http://www.geogebratube.org/student/m15410) converges to a fractal curve which is continous everywhere but nowhere differentiable.

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• Prove that the pointwise limit function of $f_j$ is continous everywhere.
• Prove that the pointwise limit function of $f_j$ is nowhere differentiable.
• Prove that the pointwise limit function of $f_j$ has infinite arc length.
(Hint: analyze $f_j$ at special nodes on the $x$-axis as marked in the animation)

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