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Statement

Let \(W(x)=\sum_{n=0}^{\infty} a^n \cos(b^n \pi x)\) with \(0<a<1\) and \(b\) a positive odd integer. Then:

  1. \(W\) is continuous on all of \(\mathbb{R}\).
  2. \(W\) is differentiable nowhere provided \(ab>1+\tfrac{3\pi}{2}\) (Weierstrass's original condition). Hardy (1916) showed non-differentiability holds under the weaker condition \(ab\ge 1\) (with \(0<a<1\), \(b>1\)).

Assumptions

  1. \(0<a<1\) ensures the amplitude series \(\sum a^n\) converges (geometric).
  2. Weierstrass's construction takes \(b\) an odd integer and couples \(a,b\) via \(ab>1+\tfrac{3\pi}{2}\); Hardy removed the integer/odd restriction and weakened the coupling to \(ab\ge1\).

Proof

  1. [verified] (Continuity via the Weierstrass M-test.) Each term satisfies \(|a^n\cos(b^n\pi x)|\le a^n=:M_n\), and \(\sum_{n\ge0}M_n= \frac{1}{1-a}<\infty\) since \(0<a<1\). By the M-test the series converges uniformly; a uniform limit of continuous functions is continuous, so \(W\in C(\mathbb{R})\).
  2. [asserted] (Nowhere differentiability, Weierstrass 1872.) Under \(ab>1+\tfrac{3\pi}{2}\) and \(b\) odd, for every \(x_0\) one constructs two sequences of points approaching \(x_0\) from opposite sides along which the difference quotients grow without bound with opposite signs, so no finite derivative can exist. The threshold constant is \(1+\tfrac{3\pi}{2}=5.71238898038469\).
  3. [asserted] (Hardy's sharpening, 1916.) Hardy proved that \(0<a<1,\ b>1,\ ab\ge1\) already forces non-differentiability everywhere, dropping both the oddness of \(b\) and the larger constant.

Verification

  • Symbolic (SymPy): the Weierstrass threshold constant evaluates to \(1+\tfrac{3\pi}{2}=5.71238898038469\) (confirmed).
  • Verified reasoning: Step 1 (continuity) is fully verified — the M-test bound \(M_n=a^n\) with \(\sum a^n=1/(1-a)<\infty\) is elementary and machine- confirmable.
  • Asserted: Steps 2-3 (nowhere differentiability) are the conclusions of Weierstrass (1872) and Hardy (1916); they are cited, not re-derived here. No finite computation can establish "no derivative exists at any point."

Historical source

Karl Weierstrass presented the function to the Royal Prussian Academy of Sciences in Berlin in 1872 as the first published example of a continuous nowhere-differentiable function. G. H. Hardy gave the definitive analysis and weakened the hypothesis to \(ab\ge1\) in 1916.

Related concepts

Citations

  1. Weierstrass, K. "Uber continuirliche Functionen eines reellen Arguments, die fur keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen." Read to the Berlin Academy, 18 July 1872.
  2. Hardy, G. H. "Weierstrass's non-differentiable function." Transactions of the American Mathematical Society 17 (1916): 301-325.