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).
Read more about this topic: Weierstrass Function
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