Let the distribution function $CDF(X,t)$ of a random variable $X$ be defined as $0$ for $ t <0, \text{Cantor function}(t)$ for $t \ge 0$ and $ t \le 1, 1$ for $ t > 1$ (for example, see http://en.wikipedia.org/wiki/Cantor_function for the used definition). What is the characteristic function of $X$? It would be an example of the characteristic function of a singular distribution.
This should probably be a comment but I'm 9 points short. The answer's on wikepedia. It's $e^{\tfrac{it}2} \prod_{i=1}^\infty cos\left(\frac t{3i}\right)$. http://en.wikipedia.org/wiki/Cantor_distribution I'm pretty sure there's no closed form of the infinite product. 

