Weierstrass function from 1872 is the function defined by
Weierstrass showed that if
- is an odd positive integer, and
then is a continuous nowhere differentiable function. Hardy proved in 1916 that one can relax the conditions on to
- , and
Here, I just want to show some graphs, hopefully providing some intuition to help understand why we expect to be non-differentiable. The idea is that the cosine terms ensure that the partial sums , though smooth, have more and more “turns” on each interval as increases, so that in the limit, has “peaks” everywhere. Below is an animation (produced using Sage) comparing the graphs of for (and ), for and , showing how the bends accumulate. (If the animations are not running, clicking on them solves the problem. As far as I can see, they do not work on mobiles.)
Paul du Bois-Reymond. Versuch einer Classiﬁcation der willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den kleinsten Intervallen, J. Reine. Angew. Math., 79, (1875), 21–37.
Hardy’s improvement appeared in