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
Famous quotes containing the word functions:
“Let us stop being afraid. Of our own thoughts, our own minds. Of madness, our own or others’. Stop being afraid of the mind itself, its astonishing functions and fandangos, its complications and simplifications, the wonderful operation of its machinery—more wonderful because it is not machinery at all or predictable.”
—Kate Millett (b. 1934)