# Weierstrass function

The **Weierstrass function** is a real-valued function defined by a trigonometric series, first presented by Karl Weierstrass in 1872, that is continuous at every real number but differentiable at none. In the classical cosine form it is written

\[ W(x) = \sum_{n=0}^{\infty} a^{n} \cos(b^{n} \pi x), \]

with \( 0 < a < 1 \) and \( b > 1 \). The series converges absolutely and uniformly, so its sum is continuous, yet the oscillations at infinitely many scales destroy the derivative everywhere. The function became the standard example of a pathological function in analysis and, through its graph, one of the first studied fractal curves.<sup>[1](https://people.math.sc.edu/girardi/m555/funkyfunctions/hardy.pdf)</sup><sup> • </sup><sup>[2](https://unseel.com/mathematics/weierstrass-function)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( W(x) = \sum_{n=0}^{\infty} a^{n} \cos(b^{n} \pi x) \), with \( 0 < a < 1 \), \( b > 1 \)<sup>[1](https://people.math.sc.edu/girardi/m555/funkyfunctions/hardy.pdf)</sup> |
| Defining property | Continuous everywhere on \( \mathbb{R} \), differentiable nowhere<sup>[1](https://people.math.sc.edu/girardi/m555/funkyfunctions/hardy.pdf)</sup> |
| First presentation | Read to the Berlin Academy on 18 July 1872; published in print in 1875 by Paul du Bois-Reymond in Crelle's Journal für die reine und angewandte Mathematik, volume 79, pages 21–37<sup>[2](https://unseel.com/mathematics/weierstrass-function)</sup> |
| Weierstrass's conditions | \( b \) an odd integer, \( 0 < a < 1 \), and \( ab > 1 + \tfrac{3}{2}\pi \)<sup>[1](https://people.math.sc.edu/girardi/m555/funkyfunctions/hardy.pdf)</sup> |
| Sharp condition | Hardy (1916) proved nowhere differentiability for \( 0 < a < 1 \), \( ab \ge 1 \), \( b > 1 \), with \( b \) not necessarily an odd integer<sup>[3](https://ltu.diva-portal.org/smash/get/diva2%3A1022983/FULLTEXT01)</sup> |
| Graph dimension | Box dimension \( D = 2 + \log \lambda / \log b \) for \( W_{\lambda,b}(x) = \sum \lambda^{n} \cos(2\pi b^{n} x) \); the Hausdorff dimension equals the same \( D \) for every integer \( b \ge 2 \) and \( \lambda \in (1/b, 1) \)<sup>[4](https://arxiv.org/abs/1309.3759)</sup><sup> • </sup><sup>[5](https://arxiv.org/abs/1505.03986)</sup> |

## Definition and continuity

The series defining \( W \) converges for every real \( x \). Since \( 0 < a < 1 \), the geometric series \( \sum a^{n} = 1/(1-a) \) converges, and \( \sup_{x} |a^{n} \cos(b^{n} \pi x)| \le a^{n} \). The Weierstrass M-test then shows that the series converges uniformly on all of \( \mathbb{R} \), and a uniform limit of continuous functions is continuous, so \( W \) is continuous everywhere.<sup>[3](https://ltu.diva-portal.org/smash/get/diva2%3A1022983/FULLTEXT01)</sup> In the common form with \( \cos(2\pi b^{n} x) \), the function is periodic with period \( 2\pi \) and is an example of a [Fourier series](https://www.edgepedia.org/fourier-series).<sup>[6](https://sites.math.washington.edu/~conroy/general/weierstrass/weier.htm)</sup>

**Nowhere differentiability.** Differentiating the series term by term would give terms of size roughly \( (ab)^{n} \), which do not tend to zero once \( ab \ge 1 \), so the formal derivative series cannot converge. Weierstrass proved non-differentiability under his original restrictions, and Hardy's 1916 paper "Weierstrass's Non-Differentiable Function" in the *Transactions of the American Mathematical Society* (volume 17, pages 301–325) extended the result to the full range \( ab \ge 1 \).<sup>[1](https://people.math.sc.edu/girardi/m555/funkyfunctions/hardy.pdf)</sup><sup> • </sup><sup>[3](https://ltu.diva-portal.org/smash/get/diva2%3A1022983/FULLTEXT01)</sup>

## History

Weierstrass read the function to the Berlin Academy on 18 July 1872 but did not publish it himself; it reached print in 1875 through Paul du Bois-Reymond's paper in Crelle's Journal, and it appears in Weierstrass's collected *Mathematische Werke* II of 1895.<sup>[2](https://unseel.com/mathematics/weierstrass-function)</sup> The example addressed a live question of the period: it was understood that a continuous function could fail to be differentiable at isolated points (as with \( |x| \) at the origin), but many texts carried incorrect arguments that a continuous function must be differentiable at some points.<sup>[7](https://api.pageplace.de/preview/DT0400.9781107595606_A24418221/preview-9781107595606_A24418221.pdf)</sup>

Earlier constructions. Around 1860, Charles Cellérier of Geneva set down the series \( \sum a^{-n} \sin(a^{n} x) \), taking \( a \) to be a sufficiently large whole number; this work only appeared in print after his death, in the *Bulletin des Sciences Mathématiques* of 1890. Bolzano's and Cellérier's examples were constructed earlier than Weierstrass's, though none is believed to have been published before it.<sup>[2](https://unseel.com/mathematics/weierstrass-function)</sup><sup> • </sup><sup>[3](https://ltu.diva-portal.org/smash/get/diva2%3A1022983/FULLTEXT01)</sup> A related function \( R(x) = \sum_{n=1}^{\infty} \sin(n^{2}x)/n^{2} \) is attributed to Riemann, who according to Kronecker and Weierstrass claimed in 1861 that it fails to be differentiable on a dense set; Hardy proved in 1916 that it has no finite derivative at any irrational and at some [rational points](https://www.edgepedia.org/rational-point), and Gerver (1970, 1971) and Smith (1972) completed the analysis of the rational cases, finding a finite derivative equal to \( 1/2 \) at points \( x = (2A+1)/(2B+1) \).<sup>[8](https://mathworld.wolfram.com/WeierstrassFunction.html)</sup>

## Regularity and the graph as a fractal

When \( ab > 1 \), writing \( a = b^{-\alpha} \) with \( 0 < \alpha < 1 \), the sum satisfies a [Hölder condition](https://www.edgepedia.org/holder-condition) of order \( \alpha \), and the local behavior is the same at every point; the case \( ab = 1 \) is more subtle, with a Hölder bound involving \( \log(1/|h|) \) and distinct classes of "rapid", "average", and "slow" points.<sup>[9](https://bibliotheque.imo.universite-paris-saclay.fr/media/filer_public/b7/3f/b73f1679-49d5-407b-bdc9-ed73a0eff4e0/k_kahane-70.pdf)</sup>

The graph of the Weierstrass function is approximately self-affine, with scales \( \lambda \) and \( 1/b \), and is regarded as the first historical example of what Benoît Mandelbrot called a fractal.<sup>[4](https://arxiv.org/abs/1309.3759)</sup><sup> • </sup><sup>[9](https://bibliotheque.imo.universite-paris-saclay.fr/media/filer_public/b7/3f/b73f1679-49d5-407b-bdc9-ed73a0eff4e0/k_kahane-70.pdf)</sup> For

\[ W_{\lambda,b}(x) = \sum_{n=0}^{\infty} \lambda^{n} \cos(2\pi b^{n} x), \qquad b \ge 2, \quad 1/b < \lambda < 1, \]

the box dimension of the graph equals \( D = 2 + \log \lambda / \log b \).<sup>[4](https://arxiv.org/abs/1309.3759)</sup> Mandelbrot conjectured in 1977 that the [Hausdorff dimension](https://www.edgepedia.org/hausdorff-dimension) equals the same \( D \). Partial results accumulated over decades: Besicovitch and Ursell gave a lower-bound estimate in 1937; Mauldin and Williams (1986) and Przytycki and Urbański (1989) proved lower bounds; Rezakhanlou (1988) and Hu and Lau (1993) settled the packing and \( K \)-dimensions; and Hunt (1998) proved the result almost surely when random phases are added.<sup>[4](https://arxiv.org/abs/1309.3759)</sup><sup> • </sup><sup>[10](https://www.mdpi.com/2504-3110/9/3/143)</sup> Barański, Bárány, and Romanowska proved in 2013 that the dimension equals \( D \) for every integer \( b \ge 2 \) when \( \lambda \) is sufficiently close to 1, above an explicit threshold \( \lambda_{b} \) tending to \( 1/\pi \) as \( b \to \infty \).<sup>[4](https://arxiv.org/abs/1309.3759)</sup> Weixiao Shen removed the restriction in 2015, proving that the Hausdorff dimension of the graph is \( D \) for every integer \( b \ge 2 \) and every \( \lambda \in (1/b, 1) \), with the same result for general non-constant \( C^{2} \) periodic \( \phi \) when \( \lambda b \) is close to 1.<sup>[5](https://arxiv.org/abs/1505.03986)</sup> For integer \( b \), the graph can be treated as an invariant repeller of an expanding dynamical system, which is how the ergodic-theoretic proofs proceed.<sup>[5](https://arxiv.org/abs/1505.03986)</sup>

## Generalizations and applications

Replacing the cosine by any non-constant periodic function \( \phi \) gives the Weierstrass-type functions \( f_{\lambda,b}^{\phi}(x) = \sum \lambda^{n} \phi(b^{n} x) \), whose graph dimension has been studied extensively; for a typical \( \mathbb{Z} \)-periodic \( C^{3} \) function \( \phi \) the Hausdorff dimension equals \( D \).<sup>[4](https://arxiv.org/abs/1309.3759)</sup> In higher dimensions, for a real analytic periodic \( \phi: \mathbb{R} \to \mathbb{R}^{d} \), the box and Hausdorff dimensions of the graph of \( W(x) = \sum \lambda^{n} \phi(b^{n} x) \) are both equal to \( \min\{\log_{\lambda^{-1}} b,\; 1 + (d - q)(1 + \log_{b} \lambda)\} \), where \( q \) is the maximum dimension of a linear space on which the projection of \( W \) is Lipschitz.<sup>[11](https://arxiv.org/abs/2404.06778)</sup>

**The Weierstrass–Mandelbrot function.** Mandelbrot extended the classical function to a two-sided series with random phases,

\[ W(t) = \sum_{n=-\infty}^{\infty} \frac{(1 - e^{i t \gamma^{n}})\, e^{i \phi_{n}}}{\gamma^{(2-D)n}}, \qquad 1 < D < 2, \quad \gamma > 1, \]

removing the largest-scale cutoff of the original; the real part is used in applications as a deterministic model of rough profiles with prescribed fractal dimension \( D \), related to the [Hurst exponent](https://www.edgepedia.org/hurst-exponent) by \( D = 2 - H \).<sup>[12](https://link.springer.com/article/10.1007/s40948-023-00532-4)</sup><sup> • </sup><sup>[13](https://pmc.ncbi.nlm.nih.gov/articles/PMC3398483/)</sup> The Weierstrass function serves as a classic model of a multifractal process and is widely used in physics and physiology; its power spectrum has the power-law form \( S(f) \sim 1/f^{\beta} \) with \( \beta = 2H + 1 \).<sup>[13](https://pmc.ncbi.nlm.nih.gov/articles/PMC3398483/)</sup> A Weierstrass–Mandelbrot drift in the [Loewner equation](https://www.edgepedia.org/loewner-equation) generates fractal curves with discrete scale invariance, and the function has been used to describe the fractal component of turbulent velocity.<sup>[14](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.82.061101)</sup><sup> • </sup><sup>[15](https://www.sciencedirect.com/science/article/pii/0169598392900596)</sup> Because the cosine is symmetric under time reversal, the classical function cannot model irreversible time series; replacing the cosines with sawtooth functions yields the asymmetric Weierstrass function, which varies the multiscale time asymmetry through a parameter \( \omega \).<sup>[13](https://pmc.ncbi.nlm.nih.gov/articles/PMC3398483/)</sup>

## References

1. [G. H. Hardy (1916). Weierstrass's Non-Differentiable Function. Transactions of the American Mathematical Society 17, 301–325.](https://people.math.sc.edu/girardi/m555/funkyfunctions/hardy.pdf)
2. [The Weierstrass Function — Continuous Everywhere, Differentiable Nowhere, Unseel](https://unseel.com/mathematics/weierstrass-function)
3. [Continuous Nowhere Differentiable Functions, MSc thesis, Luleå University of Technology](https://ltu.diva-portal.org/smash/get/diva2%3A1022983/FULLTEXT01)
4. [K. Barański, B. Bárány, J. Romanowska (2013). On the dimension of the graph of the classical Weierstrass function. arXiv:1309.3759](https://arxiv.org/abs/1309.3759)
5. [Weixiao Shen (2015). Hausdorff dimension of the graphs of the classical Weierstrass functions. arXiv:1505.03986](https://arxiv.org/abs/1505.03986)
6. [Weierstrass functions, R. Conroy, University of Washington](https://sites.math.washington.edu/~conroy/general/weierstrass/weier.htm)
7. [Fourier Analysis (Körner), preview, Cambridge University Press](https://api.pageplace.de/preview/DT0400.9781107595606_A24418221/preview-9781107595606_A24418221.pdf)
8. [Weierstrass Function, Wolfram MathWorld](https://mathworld.wolfram.com/WeierstrassFunction.html)
9. [J.-P. Kahane, Some Random Series of Functions (chapter on Riemann and Weierstrass series), Université Paris-Saclay library](https://bibliotheque.imo.universite-paris-saclay.fr/media/filer_public/b7/3f/b73f1679-49d5-407b-bdc9-ed73a0eff4e0/k_kahane-70.pdf)
10. [Progress on Fractal Dimensions of the Weierstrass Function and Weierstrass-Type Functions, Fractal and Fractional (MDPI)](https://www.mdpi.com/2504-3110/9/3/143)
11. [The high-dimensional Weierstrass functions, arXiv:2404.06778](https://arxiv.org/abs/2404.06778)
12. [Morphology differences between fractional Brownian motion and the Weierstrass-Mandelbrot function, Geomechanics and Geophysics for Geo-Energy and Geo-Resources](https://link.springer.com/article/10.1007/s40948-023-00532-4)
13. [Generating Signals with Multiscale Time Irreversibility: The Asymmetric Weierstrass Function, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC3398483/)
14. [Discrete scale invariance and stochastic Loewner evolution, Physical Review E 82, 061101 (2010)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.82.061101)
15. [On the use of the Weierstrass-Mandelbrot function to describe the fractal component of turbulent velocity, Flow Turbulence and Combustion](https://www.sciencedirect.com/science/article/pii/0169598392900596)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Real analysis*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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