dilations ; since positive linear combinations and point-wise limits of subadditive functions are subadditive, the Takagi function is subadditive for any value of the parameter . The special case of the parabola For , one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes. Differentiability For values of the parameter the Takagi function is differentiable in classical sense at any which is not a dyadic rational. Precisely, by derivation under the sign of series, for any non dyadic rational one finds where is the sequence of binary digits in the base 2 expansion of , that is, . Moreover, for these values of the function is Lipschitz of constant . In particular for the special value one finds, for any non dyadic rational , according with the mentioned For the blancmange function it is of bounded variation on no non-empty open set; it is not even locally Lipschitz, but it is quasi-Lipschitz, indeed, it admits the function as a modulus of continuity . Fourier series expansion The Takagi-Landsberg function admits an absolutely convergent Fourier series expansion: with and, for where is the maximum power of that divides . Indeed, the above triangle wave has an absolutely convergent Fourier series expansion By absolute convergence, one can reorder the corresponding double series for : putting yields the above Fourier series for Self similarity The recursive definition allows the monoid of self-symmetries of the curve to be given. This monoid is given by two generators, g and r, which act on the curve (restricted to the unit interval) as and . A general element of the monoid then has the form for some integers This acts on the curve as a linear function: for some constants a, b and c. Because the action is linear, it can be described in terms of a vector space, with the vector space basis: In this representation, the action of g and r are given by and That is, the action of a general element maps the blancmange curve on the unit interval [0,1] to a sub-interval for some integers m, n, p. The mapping is given exactly by where the values of a, b and c can be obtained directly by multiplying out the above matrices. That is: Note that is immediate. The monoid generated by g and r is sometimes called the dyadic monoid; it is a sub-monoid of