Infinitely Wrinkly Functions Having No Rate of Change at Any Moment
It would be natural to think that every function has a rate of change at all but a few moments. This, however, is not true. The Blancmange
Function and the Weierstrass
Function do not have a rate of change at a moment for any values of their independent variables. They are famous examples of functions that are never locally linear at any value of their independent variables.
The Blancmange and Weierstrass functions are like the sound wave function of Section 4.7, except that both are what we will call “infinitely wrinkly”, meaning that no matter how much you zoom in on their graphs, they remain just as wrinkly as before zooming. A consequence of these functions being infinitely wrinkly is that for any value of their independent variable, you cannot ever zoom in far enough that the function behaves linearly around that value. Therefore, there is no interval of any size around any value of their independent variables over which these functions have a rate of change that is essentially constant.
The animations in Figure 4.8.1 are approximations of the Blancmange and Weierstrass functions. The actual function definitions are in terms of the value of N becoming infinitely large. For these animations, N is conveniently small. But the animations nevertheless convey what it means for a function’s graph to be just as wrinkly after zooming as it was before. Neither function ever comes close to having a constant rate of change over any interval, no matter how small the interval becomes.
Figure 4.8.1. Approximations to the
Blancmange and Weierstrass functions. No matter how many times you zoom,
their graphs are as wrinkly as before zooming.
Exercise Set 4.8
1. The animation below shows the first eight steps of the process by which the Blancmange function is made. As a visual aid to understanding the process, the animation also shows the process being undone. Imagine that this process continued indefinitely. How does the process of making the Blancmange function explain why it has no rate of change at any moment? (Section
4.6 is highly related to this question.)
2. Define the Weierstrass functions in GC. Press ctrl-shift-S to create a summation. Zoom in repeatedly on each graph.
a) Examine the definition of the Weierstrass function. What about it makes its graph so wrinkly? (Section
4.7 is highly related to this question.)
b) The animations in Figure 4.8.1 focused on a particular value of x. Reset your graph from Part (a). Zoom in repeatedly around another value of x. Then reset the graph and try another point. Skip around the graph which point you expand around. Does it make a difference regarding which point you choose as to whether this function has a rate of change at any moment?