What's Special About This Number? Weierstrass functions. Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere".
Here is an example of one: It is not hard to show that this series converges for all x. In fact, it is absolutely convergent. It is also an example of a fourier series, a very important and fun type of series. It can be shown that the function is continuous everywhere, yet is differentiable at no values of x.