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By construction, functions of Takagi power class are similar to Takagi's continuous nowhere differentiable function. These functions have one real parameter $p>0$. They are defined by the series $S_p(x) = \sum_{n=0}^\infty…

The Takagi function is a continuous non-differentiable function on [0,1] introduced by Teiji Takagi in 1903. It has since appeared in a surprising number of different mathematical contexts, including mathematical analysis, probability…

The Takagi function $T:[0,1]\to \mathbb{R}$ is a classical example of a continuous nowhere differentiable function. In this paper, we study the discrete dynamical system generated by the Takagi function. First, we prove that for almost…

This paper examines level sets of functions of the form $f(x)=\sum_{n=0}^\infty \frac{r_n}{2^n}\phi(2^n x)$, where phi(x) is the distance from x to the nearest integer, and r_n equals 1 or -1 for each n. Such functions are referred to as…

We consider a generalized version of the Takagi function, which is one of the most famous example of nowhere differentiable continuous functions. We investigate a set of conditions to describe the rate of convergence of Takagi class…

We consider Gomory and Johnson's infinite group model with a single row. Valid inequalities for this model are expressed by valid functions and it has been recently shown that any valid function is dominated by some nonnegative valid…

We consider a one-parameter family of functions $\{F(t,x)\}_{t}$ on $[0,1]$ and partial derivatives $\partial_{t}^{k} F(t, x)$ with respect to the parameter $t$. Each function of the class is defined by a certain pair of two square matrices…

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local…

We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function.…

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,…

The Takagi function {\tau} : [0, 1] \rightarrow [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. This paper studies the level sets L(y) = {x : {\tau}(x) = y} of the Takagi function {\tau}(x). It shows that…

The Takagi function is a classical example of a continuous nowhere differentiable function. It has empty subdifferential except in a countable set where its subdifferential is $\mathbb{R}$. In this paper we characterize its…

The Takagi function is a classical example of a continuous nowhere differentiable function. In this paper we prove that it is nowhere approximately derivable.

For any integer $b\geq2$ and real series $\{c_n\}$ such that $\sum_{n=0}^\infty|c_n|<\infty$, the generalized Takagi function $f_{{\mathbf c},b}(x)$ is defined by $$ f_{{\mathbf c},b}(x):=\sum_{n=0}^\infty c_n\phi(b^n x), \quad x\in [0,1],…

The Takagi-van der Waerden functions are a well-known class of continuous but nowhere differentiable functions. In this paper, we study their weighted versions, the Takagi-van der Waerden class functions $f_{r,a}(x)$, from a probabilistic…

In this paper, we prove that for some Generalized Takagi Classes, in particular for the Takagi-Van der Waerden Class, the functions are nowhere differentiable if, and only if, the sequence of weights does not belong to $c_0$.

Let T be Takagi's continuous but nowhere-differentiable function. Using a representation in terms of Rademacher series due to N. Kono, we give a complete characterization of those points where T has a left-sided, right-sided, or two-sided…

In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition…

This paper examines the level sets of the continuous but nowhere differentiable functions \begin{equation*} f_r(x)=\sum_{n=0}^\infty r^{-n}\phi(r^n x), \end{equation*} where $\phi(x)$ is the distance from $x$ to the nearest integer, and $r$…

For any measure preserving system $(X,\mathcal{B},\mu,T_1,\ldots,T_d),$ where we assume no commutativity on the transformations $T_i,$ $1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of…