Weierstrass Function - Density of Nowhere-differentiable Functions

Density of Nowhere-differentiable Functions

It turns out that the Weierstrass function is far from being an isolated example: although it is "pathological", it is also "typical" of continuous functions:

• In a topological sense: the set of nowhere-differentiable real-valued functions on is comeager in the vector space C(R) of all continuous real-valued functions on with the topology of uniform convergence.
• In a measure-theoretic sense: when the space C(R) is equipped with classical Wiener measure γ, the collection of functions that are differentiable at even a single point of has γ-measure zero. The same is true even if one takes finite-dimensional "slices" of C(R): the nowhere-differentiable functions form a prevalent subset of C(R).