Weierstrass function
This is my relatively-simple attempt to model something that might help visualize how the Weierstrass function is constructed. As the Wikipedia entry says, this is an example of a function that is continuous for all real x, yet is nowhere-differentiable. The fact that such a function even exists is extremely counter-intuitive.
The actual definition of the Weierstrass function is
.
In this construction, the upper Σ limit is N, controlled by a slider, (rather than ∞).
As you increase the value of N, the graph quickly becomes more "fractal-like." Drag the corners of the light-blue shaded rectangle to define the zoomed-in view at the right of the graphics region.
The Weierstrass has conditions 0<a<b, where b is a positive odd integer. Check the "requirements" checkbox to impose these restrictions on the a and b sliders. However, just to get a rough idea of how varying each parameter changes the function, it might be easier to play with the sliders while the checkbox is unchecked.