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