The elementary continuous functions we work with in calculus tend to be relatively well-behaved, in the sense that they are differentiable everywhere except some isolated set of points. The prototypical example is the map , which fails to be differentiable at zero since for , so does not exist. More generally, let and with

– that is, the slope of the line segment joining and differs from the slope of the line segment joining and . Define such that and the graph of is

That is, is a piecewise linear function connecting the points – like the functions used in the previous post. Then is differentiable on . Moreover, if is a strictly increasing sequence with satisfying the aforementioned slope assumption, then defining as above, we see that is continuous on and differentiable at all but countably many points.

The paths of Brownian motion are even more pathological – they are with probability differentiable nowhere! Let be a standard Brownian motion and

We will show that . For each integer and , set

and

Suppose , then since

exists, we may choose , such that

Let be a positive integer with , and put

Then

so that

and similarly

It follows that

Now, the increments are i.i.d. with distribution, so

from which we have

It follows that

Let , then so . Fatou’s lemma then yields

from which we conclude.

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