Continuous functions that are not differentiable

As mentioned on Friday, the hypothesis in the Mean Value Theorem (and Rolle's Theorem) that f(x) is continuous on [a,b] and differentiable on (a,b) covers most nice functions that we deal with in class. There is one family of examples that we've dealt with a lot that have at least one point of discontinuity:

f(x) = |x-c|, where c shifts the function so that the cusp/point lands in [a,b].

What are some variants of this function? (Think of a piecewise function that looks looks like a sawtooth--lots of little absolute value functions put together. How would you define this function?)

Now, some pathological examples. The Weierstrass function (here at wikipedia) is one very interesting example of a function that is *continuous everywhere* (why?) but differentiable nowhere (a bit harder to prove). The discussion on Wikipedia is a little beyond what we've covered in class, but from the graph you get the idea that it essentially builds on the "jagged" idea of the absolute function. Weierstrass found a way to make a function that changes direction infinitely often.

Another example is due to Bolzano. Here is a link at Math World (here and a definition of the function here). The idea is similar, but Bolzano did not use continuous cosine functions in the construction. He simply extended the sawtooth idea (ad infinitum).

-Jesse