Definition:Weierstrass Function

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Definition

Let $a \in \openint 0 1$.

Let $b$ be a strictly positive odd integer such that:

$\ds a b > 1 + \frac 3 2 \pi$

Let $f: \R \to \R$ be a real function defined by:

$\ds \map f x = \sum_{n \mathop = 0}^\infty a^n \map \cos {b^n \pi x}$

for each $x \in \R$.


We call $f$ a Weierstrass function.


Also see

  • Results about the Weierstrass function can be found here.


Historical Note

Karl Weierstrass first discussed a real function which was continuous everywhere but differentiable nowhere in his lectures in $1861$.

The explicit construction given here did not appear in print until one of his students published it (with Weierstrass's permission) in $1874$.


Sources