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
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($\text {1815}$ – $\text {1897}$)