Real Analysis Exchange

The Takagi Function: a Survey

Pieter C. Allaart and Kiko Kawamura

Full-text: Open access


This paper sketches the history of the Takagi function T and surveys known properties of T, including its nowhere-differentiability, modulus of continuity, graphical properties and level sets. Several generalizations of the Takagi function, in as far as they are based on the tent map, are also discussed. The final section reviews a number of applications of the Takagi function to various areas of mathematics, including number theory, combinatorics and classical real analysis.

Article information

Real Anal. Exchange, Volume 37, Number 1 (2011), 1-54.

First available in Project Euclid: 30 April 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
Secondary: 26A16: Lipschitz (Hölder) classes 26A78 26A80

Takagi function van der Waerden function continuous nowhere differentiable function modulus of continuity infinite derivatives level set Hausdorff dimension Lebesgue's singular function functional equation


Allaart, Pieter C.; Kawamura, Kiko. The Takagi Function: a Survey. Real Anal. Exchange 37 (2011), no. 1, 1--54.

Export citation


  • S. Abbott, J. M. Anderson and L. D. Pitt, Slow points for functions in the Zygmund class $\Lambda_d^*$, Real Anal. Exchange 32 (2006/07), no. 1, 145–170.
  • P. C. Allaart, Distribution of the extrema of random Takagi functions, Acta Math. Hungar. 121 (2008), no. 3, 243–275.
  • P. C. Allaart, On a flexible class of continuous functions with uniform local structure, J. Math. Soc. Japan 61 (2009), no. 1, 237–262.
  • P. C. Allaart, An inequality for sums of binary digits, with application to Takagi functions, J. Math. Anal. Appl. 381 (2011), no. 2, 689–694.
  • P. C. Allaart, How large are the level sets of the Takagi function? Preprint, arXiv:1102.1616 (2011).
  • P. C. Allaart, On the distribution of the cardinalities of level sets of the Takagi function, preprint, arXiv:1107.0712 (2011).
  • P. C. Allaart and K. Kawamura, Extreme values of some continuous, nowhere differentiable functions, Math. Proc. Camb. Phil. Soc. 140 (2006), no. 2, 269–295.
  • P. C. Allaart and K. Kawamura, The improper infinite derivatives of Takagi's nowhere-differentiable function, J. Math. Anal. Appl. 372 (2010), no. 2, 656–665.
  • E. de Amo, I. Bhouri, M. Díaz Carrillo, and J. Fernández-Sánchez, The Hausdorff dimension of the level sets of Takagi's function, Nonlinear Anal. 74 (2011), no. 15, 5081–5087.
  • E. de Amo, M. Díaz Carrillo, and J. Fernández-Sánchez, Singular functions with applications to fractals and generalized Takagi functions, preprint (2011).
  • E. de Amo and J. Fernández-Sánchez, Takagi's function revisited from an arithmetical point of view, Int. J. Pure Appl. Math. 54 (2009), no. 3, 407–427.
  • J. M. Anderson and L. D. Pitt, Probabilistic behavior of functions in the Zygmund spaces $\Lambda^*$ and $\lambda^*$, Proc. London Math. Soc. 59 (1989), no. 3, 558-592.
  • Y. Baba, On maxima of Takagi-van der Waerden functions, Proc. Amer. Math. Soc. 91 (1984), no. 3, 373–376.
  • R. Balasubramanian, S. Kanemitsu and M. Yoshimoto, Euler products, Farey series, and the Riemann hypothesis. II, Publ. Math. Debrecen 69 (2006), no. 1-2, 1–16.
  • E. G. Begle and W. L. Ayres, On Hildebrandt's example of a function without a finite derivative, Amer. Math. Monthly 43 (1936), no. 5, 294–296.
  • A. S. Besicovitch and H. D. Ursell, Sets of fractional dimensions (V). On dimensional numbers of some continuous curves, J. London Math. Soc. 12 (1937), 18–25.
  • P. Billingsley, Van Der Waerden's Continuous Nowhere Differentiable Function, Amer. Math. Monthly 89 (1982), no. 9, 691.
  • Z. Boros, An inequality for the Takagi function. Math. Inequal. Appl. 11 (2008), no. 4, 757–765.
  • J. B. Brown and G. Kozlowski, Smooth interpolation, Hölder continuity, and the Takagi-van der Waerden function. Amer. Math. Monthly 110 (2003), no. 2, 142–147.
  • Z. Buczolich, Micro tangent sets of continuous functions, Math. Bohem. 128 (2003), 147–167.
  • Z. Buczolich, Irregular 1-sets on the graphs of continuous functions. Acta Math. Hungar. 121 (2008), no. 4, 371–393.
  • F. S. Cater, On van der Waerden's nowhere differentiable function. Amer. Math. Monthly 91 (1984), no. 5, 307–308.
  • J. Coquet, Power sums of digital sums J. Number Theory 22 (1986), 161–176.
  • W. F. Darsow, M. J. Frank and H.-H. Kairies, Errata: “Functional equations for a function of van der Waerden type", Rad. Mat. 5 (1989), no. 1, 179–180.
  • H. Delange, Sur la fonction sommatoire de la fonction “somme des chiffres", Enseignement Math. 21 (1975), 31–47.
  • G. Faber, Einfaches Beispiel einer stetigen nirgends differenzierbaren Funktion, Jahresber. Deutschen Math.-Verein 16 (1907), 538–540.
  • J. Franel, Les suites de Farey et les problémes des nombres premiers, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl (1924), 198–201.
  • P. Frankl, M. Matsumoto, I. Z. Ruzsa and N. Tokushige, Minimum shadows in uniform hypergraphs and a generalization of the Takagi function. J. Combin. Theory Ser. A 69 (1995), no. 1, 125–148.
  • N. G. Gamkrelidze, On a probabilistic properties of Takagi's function (sic), J. Math. Kyoto Univ. 30 (1990), no. 2, 227–229.
  • C. J. Guu, The McFunction [The Takagi function]. Selected topics in discrete mathematics (Warsaw, 1996). Discrete Math. 213 (2000), no. 1-3, 163–167.
  • M. Hata and M. Yamaguti, Takagi function and its generalization, Japan J. Appl. Math. 1 (1984), 183–199.
  • A. Házy and Zs. Páles, On approximately midconvex functions, Bull. London Math. Soc. 36 (2004), 339–350.
  • T. H. Hildebrandt, A simple continuous function with a finite derivative at no point, Amer. Math. Monthly 40 (1933), no. 9, 547–548.
  • J.-P. Kahane, Sur l'exemple, donné par M. de Rham, d'une fonction continue sans dérivée, Enseignement Math. 5 (1959), 53–57.
  • H.-H. Kairies, Takagi's function and its functional equations, Rocznik Nauk.-Dydakt. Prace Mat. 15 (1998), 73–83.
  • H.-H. Kairies, W. F. Darsow and M. J. Frank, Functional equations for a function of van der Waerden type. Rad. Mat. 4 (1988), no. 2, 361–374.
  • N. I. Katzourakis, A Hölder continuous nowhere improvable function with derivative singular distribution, preprint, (2011).
  • K. Kawamura, On the classification of self-similar sets determined by two contractions on the plane, J. Math. Kyoto Univ. 42 (2002), 255–286.
  • K. Kawamura, On the set of points where Lebesgue's singular function has the derivative zero, Proc. Japan Acad. Ser. A 87 (2011), 162–166.
  • K. Knopp, Ein einfaches Verfahren zur Bildung stetiger nirgends differenzierbarer Funktionen. Math. Zeitschr. 2 (1918), 1–26.
  • D. E. Knuth, The art of computer programming, Vol. 4, Fasc. 3, Addison-Wesley: Upper Saddle River, NJ, 2005.
  • Z. Kobayashi, Digital sum problems for the Gray code representation of natural numbers, Interdiscip. Inform. Sci. 8 (2002), 167–175.
  • Z. Kobayashi, T. Okada, T. Sekiguchi and Y. Shiota, Applications of measure theory to digital sum problems, Sem. Math. Sci., Keio Univ. 35 (2006), 95–109.
  • N. Kôno, On generalized Takagi functions, Acta Math. Hungar. 49 (1987), 315–324.
  • M. Krüppel, On the extrema and the improper derivatives of Takagi's continuous nowhere differentiable function, Rostock. Math. Kolloq. 62 (2007), 41–59.
  • M. Krüppel, Takagi's continuous nowhere differentiable function and binary digital sums, Rostock. Math. Kolloq. 63 (2008), 37–54.
  • M. Krüppel, On the improper derivatives of Takagi's continuous nowhere differentiable function, Rostock. Math. Kolloq. 65 (2010), 3–13
  • S. T. Kuroda, Private communication.
  • J. C. Lagarias, The Takagi function and its properties, preprint.
  • J. C. Lagarias and Z. Maddock, Level sets of the Takagi function: local level sets, arXiv:1009.0855 (2010).
  • J. C. Lagarias and Z. Maddock, Level sets of the Takagi function: generic level sets, arXiv:1011.3183 (2010).
  • F. Ledrappier, On the dimension of some graphs, Contemp. Math. 135 (1992), 285–293.
  • J. S. Lipinski, On zeros of a continuous nowhere differentiable function, Amer. Math. Monthly 73 (1966), no. 2, 166–168.
  • Z. Lomnicki and S. Ulam\@, Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités I. Variables indépendantes. Fund. Math. 23 (1934), 237–278.
  • Z. Maddock, Level sets of the Takagi function: Hausdorff dimension, Monatsh. Math. 160 (2010), no. 2, 167–186.
  • J. Makó and Zs. Páles, Approximate convexity of Takagi type functions, J. Math. Anal. Appl. 369 (2010), 545–554.
  • B. Martynov, On maxima of the van der Waerden function, Kvant (1982) June, 8–14 (in Russian).
  • B. Martynov, Van der Waerden's pathological function: examining a “miserable sore". Quantum 8 (1998), no. 6, 12–19.
  • R. Mauldin and S. Williams, On the Hausdorff dimension of some graphs, Trans. Amer. Math. Soc. 298 (1986), 793–803.
  • K. Odani, On Takagi's nowhere differentiable function, Sugaku 47 (1995), 422-423 (in Japanese)
  • T. Okada, T. Sekiguchi and Y. Shiota, An explicit formula of the exponential sums of digital sums, Japan J. Indust. Appl. Math. 12 (1995), 425–438.
  • H. Okamoto, The evolution of the function concept (in Japanese), Kindai Kagakusha, in press.
  • G. de Rham, Sur un exemple de fonction continue sans dérivée. Enseignement Math. 3 (1957), 71–72.
  • G. de Rham, Sur quelques courbes definies par des equations fonctionnelles, Rend. Sem. Mat. Torino 16 (1957), 101–113.
  • R. Salem, On some singular monotonic functions which are strictly increaing, Trans. Amer. Math. Soc. 53 (1943), 427–439.
  • C. D. Savage, A survey of combinatorial Gray codes, SIAM Rev. 39 (1997), 605–629.
  • S. R. Schubert, On A Function of Van Der Waerden. Amer. Math. Monthly 70 (1963), no. 4, 402.
  • T. Sekiguchi and Y. Shiota, A generalization of Hata-Yamaguti's results on the Takagi function. Japan J. Appl. Math. 8 (1991), 203–219.
  • A. Shidfar and K. Sabetfakhri, On the Continuity of Van Der Waerden's Function in the Hölder Sense, Amer. Math. Monthly 93 (1986), no. 5, 375–376.
  • A. Shidfar and K. Sabetfakhri, On the Hölder continuity of certain functions, Exposition. Math. 8 (1990), 365–369.
  • B. Solomyak, On the random series $\sum\pm\lambda^n$ (an Erdős problem), Ann. Math. (2) 142 (1995), no. 3, 611–625.
  • K. B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math. 32 (1977), no. 4, 713–730.
  • H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics, preprint, (2010).
  • J. Tabor and J. Tabor, Generalized approximate midconvexity, Control Cybernet. 38 (2009), no. 3, 655–669.
  • J. Tabor and J. Tabor, Takagi functions and approximate midconvexity, J. Math. Anal. Appl. 356 (2009), no. 2, 729–737.
  • T. Takagi, A simple example of the continuous function without derivative, Proc. Phys.-Math. Soc. Japan 1 (1903), 176-177. The Collected Papers of Teiji Takagi, S. Kuroda, Ed., Iwanami (1973), 5–6.
  • R. Tambs-Lyche, Une fonction continue sans dérivée, Enseignement Math. 38 (1939/40), 208–211.
  • J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 (1968), 21–25.
  • Y. Tsujii, On modified Takagi functions of two variables, J. Math. Kyoto Univ. 25 (1985), no. 3, 577–581.
  • B. W. van der Waerden, Ein einfaches Beispiel einer nicht-differenzierbaren stetigen Funktion, Math. Z. 32 (1930), 474–475.
  • Y. Yamaguchi, K. Tanikawa and N. Mishima, Fractal basin boundary in dynamical systems and the Weierstrass-Takagi functions, Phys. Lett. A 128 (1988), no. 9, 470–478.
  • A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47–76.