Definition:Weierstrass Function/Historical Note

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Historical Note on Weierstrass Function

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

The function he initially demonstrated was defined as the sum of a Fourier Series:

$\displaystyle f \left({x}\right) = \sum_{n \mathop \ge 0} a^n \cos \left({b^n \pi x}\right)$

where:

$0 < a < 1$
$b$ is a (strictly) positive odd integer

such that:

$a b > 1 + \dfrac 3 2 \pi$

It did not appear in print until one of his students published it (with Weierstrass's permission) in $1874$.


Sources