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

[list][br][*] Prove that the pointwise limit function of [math]f_j[/math] is continous everywhere.[br][*] Prove that the pointwise limit function of [math]f_j[/math] is nowhere differentiable.[br][*] Prove that the pointwise limit function of [math]f_j[/math] has infinite arc length.[br](Hint: analyze [math]f_j[/math] at special nodes on the [math]x[/math]-axis as marked in the animation)[br][/list]