Example 8.3.4 (d): Weierstrass Function is continuous
This function is part of a group of functions collectively known as Weierstrass functions. All of them are defined as a sum of trig functions, they are continuous everywhere, but not differentiable anywhere.. Here are a few representatives.
(3/4)n| sin(4n x) | (3/4)n cos(4n x) (1/2)n sin(2n x)
For each function you should zoom in by dragging the mouse. You will see that no matter how often you zoom in, you will see the graph looks continuous, but has any number of "spikes". Geometrically that implies continuity everywhere but differentiability nowhere. Since such functions are "strange" they are sometimes called "Weierstrass Monster" functions.
To formally prove continuity is straight-forward, now that we know Weierstrass' theorem: let
fn(x) = (3/4)n| sin(4n x) |
Then fn(x) is continuous and the sup-norm is:
|| fn || = (3/4)n
But now the Weierstrass convergence theorem applies directly to show that
(3/4)n| sin(4n x) |
converges absolutely and uniformly, and is continuous. Other Weierstrass functions are handled in a similar way - the details are left to the reader. It is much harder to show that given any x, the function is not differentiable at that x. We will prove that later.
The IRA logo, in fact, was created by plotting several partial sums of fn(x) = (1/2)n| sin(2n x) | and by coloring in some areas: