The Weierstrass Function

What is it useful for: are these things just pathologies? (Density of such objects: continuous, but not diff anywhere, in the set $C[0,1]$ with Weiner measure. Work out what this is, and why it's a natural fit for such functions. How many such functions? Defined by values on a countable set: given $f:Q\rightarrow Q$ continuous it extends. That is, if you have a function in $C[0,1]$ it is in 1-1 correspondence with a function on a countable subset of its domain. So... no more than the number of sequences of real numbers, which is c.