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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 has a relatively small value. But the animations nevertheless convey what it means for a functionâ€™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.
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