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 $f_j$ 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)