Since Weierstrass it has been known that there are functions that are continuous but nowhere differentiable.  A beautiful example (with probability 1) is any Brownian path.  Brownian paths can be constructed either in space, via Brownian bridge, or in Fourier space, via random Fourier series.

What about functions, which we call "smoothies", that are $C^\infty$ but nowhere analytic?  This case is less familiar but analogous, and again one can do the construction either in space or Fourier space.  We present the ideas and illustrate them with the new Chebfun $\tt{smoothie}$ command.  In the complex plane, the same idea gives functions analytic in the open unit disk and $C^\infty$ on the unit circle, which is a natural boundary.