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.

The main goal of this note is to prove the following theorem. If \(A_n\) is a sequence of measurable sets in a \(\sigma\)-finite measure space \((X, \mathcal{A}, \mu)\) that covers \(\mu\)-a.e. \(x \in X\) infinitely many times, then there exists a sequence of integers \(n_i\) of density zero so that \(A_{n_i}\) still covers \(\mu\)-a.e. \(x \in X\) infinitely many times. The proof is a probabilistic construction. As an application we give a simple direct proof of the known theorem that the ideal of density zero subsets of the natural numbers is random-indestructible, that is, random forcing does not add a co-infinite set of naturals that almost contains every ground model density zero set. This answers a question of B. Farkas.

In this paper we consider the properties of a family of probability (continuous and singular) measures, which will be called Binomial measures because of their relationship with the binomial model in probability. These measures arise in many applications with different notations. Many properties in common with Lebesgue measure hold true for this family, sometimes unexpectedly.

In this paper we introduce and study a certain zeta function and its zeros for conformal graph directed Markov systems (GDMS). These zeros are referred to as resonances. We specify a list of geometric, combinatoric and analytic conditions on the GDMS under which this zeta function is indeed well defined and even holomorphic on the whole complex plane. In addition, we prove that there is a half-plane where there are no zeros. Finally, we transfer a result of Guillopé et al. in \cite{GuLiZw} on the zeros of the Selberg zeta function to our setting. More precisely, we give an upper bound for the number of resonances in a strip in terms of the Hausdorff dimension of the limit set of the GDMS. We also briefly discuss relations to other zeta functions, in particular to the Selberg zeta function associated to a Kleinian group of Schottky type and to the geometric zeta function associated to a fractal string. Since the definition of the zeta function introduced in our paper is based on the transfer operator associated to the GDMS, these relations to other zeta functions indicate that our zeta function is a natural generalization of these zeta functions to conformal GDMSs.

In this note a theorem to determine the box dimension of the graph of the limit of a sequence of α-Hölder functions is established. By application of such a theorem the box dimensions of the graphs of some functions that are generalizations of Weierstrass-type functions are determined.

In this note we introduce three problems related to the topic of finite Hausdorff moments. Generally speaking, given the first \(n+1\) (\(n\in \mathbb N\cup \{0\}\)) moments, \(\alpha_0\), \(\alpha_1\),..., \(\alpha_n\), of a real-valued continuously differentiable function \(f\) defined on \([0,1]\), what can be said about the size of the image of \(\frac{df}{dx}\)? We make the questions more precise and we give answers in the cases of three or fewer moments and in some cases for four moments. In the general situation of \(n+1\) moments, we show that the range of the derivative should contain the convex hull of a set of \(n\) numbers calculated in terms of the Bernstein polynomials, \(x^k(1-x)^{n+1-k}\), \(k=1,2,...,n\), which turn out to involve expressions just in terms of the given moments \(\alpha_i\), \(i=0,1,2,...n\). In the end we make some conjectures about what may be true in terms of the sharpness of the interval range mentioned before.

Using Schoenberg's theory of variation-diminishing integral operators of convolution type variation diminishing wavelets and wavelets of specific changes in sign are constructed. An inversion formula involving derivatives of the wavelet transform is established. Wavelets generated by Tanno's form of convolution kernels and \(H\)-functions are also investigated. Results are illustrated by means of examples and figures.

The continuity of \(f(x,y)\) at \((x_0,y_0)\) can be described by the behavior of \(f\) along a collection of paths toward \((x_0,y_0)\) if the collection is rich enough. The collection of paths that are \(\mathcal{C}^1\) and convex is rich enough but the collection of differentiable functions with bounded derivatives is not. The collection of \(\mathcal{C}^n\) parameterized paths \((x(t),y(t))\) for any \(n\gt 0\) is also rich enough to capture continuity.

In this note we discuss some interconnections between the space \(BV_p[a,b]\) (\(1\leq p\lt\infty\)) of functions of bounded \(p\)-variation (in Wiener's sense) and the space \(Lip_\alpha[a,b]\) (\(0\lt\alpha\leq 1\)) of Hölder continuous functions. In particular, we show that \(f\in BV_p[a,b]\) if and only if \(f=g\circ \tau\), with \(g\in Lip_{1/p}[a,b]\) and \(\tau\) being monotone, and that \(f\in BV_p[a,b] \cap C[a,b]\) if and only if \(f=g\circ \tau\), with \(g\in Lip_{1/p}[a,b]\) and \(\tau\) being a homeomorphism.

The least squares derivative and the approximate derivative are both generalizations of the ordinary derivative. The existence of either of these generalized derivatives does not guarantee the existence of the other and it is even possible for both generalized derivatives to exist at a point but have different values. Several examples of such functions are presented in this paper. In addition, conditions for which the existence of the approximate derivative implies the existence (and equality) of the least squares derivative are stated and proved. These conditions involve the notion of Hölder continuity and a stronger version of approximate differentiability.

In this paper we construct a differentiable function \( F : \mathbb{R} \to \mathbb{R} \) that does not have a third order symmetric Riemann derivative at any point. In fact, \[ \underline{SRD}^3F(x) = \liminf_{h \to 0} \tfrac{F(x + 3h) - 3F(x+h) + 3F(x - h) - F(x - 3h)}{(2h)^3} = - \infty \] and \[ \overline{SRD}^3F(x) = \limsup_{h \to 0 }\tfrac{F(x + 3h) - 3F(x+h) + 3F(x - h) - F(x - 3h)}{(2h)^3} = + \infty \] for every \( x \in \mathbb{R}. \)

We produce necessary and sufficient conditions for the pointwise product of two uniformly continuous real-valued functions defined on a metric space to be uniformly continuous.

If a sum of the form \[\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1})\] is used without the familiar requirement that the sequence of points \(a=x_0, x_1, \dots, x_n=b\) is increasing, do we still get a useful approximation to the integral? With a suitable set of hypotheses the answer is yes. We give applications to change of variable formulas and the problem of characterizing derivatives.