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