Differentiable Continuous Function Research Articles

Constructing Continuous Functions

Summary This paper introduces a simple geometric construction method for graphs of continuous functions. Graphs of a nowhere differentiable continuous function, a Lipschitz function that is not differentiable on a perfect set, and a continuous function whose level sets are countable are constructed using this method. We argue that introducing students to this method early on in Real Analysis fosters proper geometric intuition about continuity and provides for simple geometric proofs of non-differentiability properties.

On the rate of convergence for Takagi class functions

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 functions from the probabilistic point of view: The law of large numbers, the central limit theorem, and the law of iterated logarithm. On the other hand, we show that the Takagi function itself does not satisfy the law of large numbers in the usual sense.

Non-differentiability and Hölder properties of self-affine functions

We consider the class of self-affine functions. Firstly, we characterize all nowhere differentiable self-affine continuous functions. Secondly, given a self-affine continuous function ϕ, we investigate its Hölder properties. We find its best uniform Hölder exponent and when ϕ is C1, we find the best uniform Hölder exponent of ϕ′. Thirdly, we show that the Hölder cut of ϕ takes the same value almost everywhere for the Lebesgue measure. This last result is a consequence of the Borel strong law of large numbers.

On the local fractional derivative of everywhere non-differentiable continuous functions on intervals

We first prove that for a continuous function f(x) defined on an open interval, the Kolvankar-Gangal’s (or equivalently Chen-Yan-Zhang’s) local fractional derivative f(α)(x) is not continuous, and then prove that it is impossible that the KG derivative f(α)(x) exists everywhere on the interval and satisfies f(α)(x) ≠ 0 in the same time. In addition, we give a criterion of the nonexistence of the local fractional derivative of everywhere non-differentiable continuous functions. Furthermore, we construct two simple nowhere differentiable continuous functions on (0, 1) and prove that they have no the local fractional derivatives everywhere.

Optimal charging strategy for a residential PEV battery considering bidirectional trade and frequency regulation

A practical charging strategy for an individual plug-in electric vehicle (PEV) is proposed. A cost function is constructed to reflect bidirectional electricity trade arising from surplus photovoltaic generation under the reward system called feed-in tariffs and time-of-use electricity pricing. It is then extended to include the revenue for providing vehicle-to-grid frequency regulation. In order to take advantages of high performance existing solvers, several mathematical techniques are applied to transform the discrete non-linear cost function to a differentiable continuous function. In the latter part of the paper, simulations are performed to verify and show the performance of the developed model. A quantitatively developed battery wear model is also applied during the simulations to calculate the effective operational cost, and to compare the costs for different control strategies. From the result of case study, the economic feasibility of the frequency regulation under the given circumstance is addressed as well.

Darboux property of Gâteaux derivatives of functions on ℝ n

D. Preiss proved that the graph of the derivative of a continuous Gâteaux-differentiable function f : ℝ2 → ℝ is always connected. We show that this is no longer true in higher dimensions: we construct a continuous, Gâteaux-differentiable function f : ℝ3 → ℝ for which the range of its gradient mapping {∇ f(x) : x ∈ ℝ3} is disconnected. We also give an example of an approximately differentiable continuous function on ℝ2 such that the range of its gradient mapping is disconnected.

Chain rules for approximate Jacobians of continuous functions

In this paper we prove a fuzzy chain rule for approximate Jacobians of continuous functions. An exact chain rule is also established by using recession approximate Jacobian matrices. The result is applied to obtain an open mapping theorem for continuous functions. In particular, a sufficient condition for openness of almost everywhere Gâteaux differentiable continuous functions is derived.

Graphs of Gâteaux Derivatives are <em>w</em>∗-Connected

We show that if $(X, \|. \|)$ is a separable Banach space, $\Omega \subset X$ is open, connected and $f: \Omega \to \mathbb{R}$ is an everywhere Gâteaux differentiable Lipschitz continuous function, then the graph of the derivative of $f$ is connected in $(\Omega,\| . \|)\times(X^\star , w^\star)$.

Construction and Dimension Analysis for a Class of Fractal Functions

In this paper, we construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, such as Bush type functions. These functions are fractal functions since their graphs are in general fractal sets. Under certain conditions, we investigate the fractal dimensions of the graphs of these functions, compute the precise values of Box and Packing dimensions, and evaluate the Hausdorff dimension. Meanwhile, the Holder continuity of such functions is also discussed.

A Nowhere Differentiable Continuous Function Constructed by Infinite Products

(2002). A Nowhere Differentiable Continuous Function Constructed by Infinite Products. The American Mathematical Monthly: Vol. 109, No. 4, pp. 378-380.

A Nowhere Differentiable Continuous Function Constructed by Infinite Products

A Nowhere Differentiable Continuous Function Constructed by Infinite Products

A Nowhere Differentiable Continuous Function Constructed Using Cantor Series

The examples of continuous nowhere differentiable functions given in most analysis texts involve the uniform convergence of a series of functions (see Hobson [1, pp. 401412]). In the last twenty years interest in this subject has been renewed ([2]-[7]). In this note we constlruct a new elementary example by using the Cantor series, which is very accessible and needs only the basic notion of limit. Let qn > 2 be a sequence of positive integers. A Cantor series, or Cantor expansion, for a real number x E [0, 1] is analogous to a decimal expansion, where numbers other than powers of ten can serve as denominators:

A Nowhere Differentiable Continuous Function

A Nowhere Differentiable Continuous Function

A Nowhere Differentiable Continuous Function

The examples of continuous nowhere differentiable functions given in most analysis texts involve the uniform convergence of a series of functions [1, pp. 401-412]. In the last twenty years this subject has again received a good deal of attention; see [2]-[6]. The purpose of this note is to give a new elementary example that needs only the basic notion of limit and is very accessible and appropriate for a first calculus course. For each real number x E [0, 1], let its base-b expansion be

Minimum shadows in uniform hypergraphs and a generalization of the Takagi function

The shadow function is closely related to the Kruskal-Katona Theorem. The Takagi function is a standard example of a nowhere differentiable continuous function. The purpose of this paper is to exhibit a rather surprising relationship between the shadow function and the Takagi function. Using this relationship, one can approximately compute the size of minimum shadows in uniform hypergraphs with a given number of edges. In order to describe the asymptotic behaviour of the size of shadows, we introduce a new, generalized Takagi function. The results explain the difficulties, often encountered when using the best possible bounds arising from the Kruskal-Katona Theorem.

Deterministic diffusion, De Rham equation and fractal eigenvectors

We study the left eigenvectors of the Frobenius-Perron operator for a piecewise linear map inducing deterministic diffusion. We show that the left eigenvectors are linear functionals associated with nowhere differentiable continuous functions satisfying De Rham's functional equation. We study moreover in some detail the fractal properties of these eigenfunctions.

The Complexity Of Nowhere Differentiable Continuous Functions

It was not always clear that there could exist a continuous function which was differentiable at no point. (Such functions are now known as nowhere differentiable continuous functions. By “differentiable” we mean having a finite derivative.) In fact in 1806 M. Ampere [2] even tried to show that no such function could exist but his reasonings were later discovered to be fallacious. Of the early attempts at constructing a nowhere differentiable continuous function mention must be made of B. Bolzano. In a manuscript dated around 1830, (see [21]) he constructed a continuous function on an interval and showed that it was not differentiable on a dense set of points. (It was later shown by K. Rychlik [21] that this function was in fact nowhere differentiable.)

The Takagi function and its generalization

In this paper, we generalize the Takagi function which is known as a nowhere differentiable continuous function, and we clarify the close relation between the Takagi function and Lebesgue’s singular function with de Rham’s functional equation.

On Maxima of Takagi-van der Waerden Functions

Generalizing Takagi’s function ${F_2}\left ( x \right )$ and van der Waerden’s function ${F_{10}}\left ( x \right )$, we introduce a class of nowhere differentiable continuous functions ${F_r}\left ( x \right )$, $r \geqslant 2$. Some properties of ${F_r}\left ( x \right )$ concerning especially maxima are discussed. When $r$ is even, the Hausdorff dimension of the set of ${x^,}$’s giving the maxima of ${F_r}\left ( x \right )$ is proved to be $1/2$.

On maxima of Takagi-van der Waerden functions

Generalizing Takagi’s function F 2 ( x ) {F_2}\left ( x \right ) and van der Waerden’s function F 10 ( x ) {F_{10}}\left ( x \right ) , we introduce a class of nowhere differentiable continuous functions F r ( x ) {F_r}\left ( x \right ) , r ⩾ 2 r \geqslant 2 . Some properties of F r ( x ) {F_r}\left ( x \right ) concerning especially maxima are discussed. When r r is even, the Hausdorff dimension of the set of x , {x^,} ’s giving the maxima of F r ( x ) {F_r}\left ( x \right ) is proved to be 1 / 2 1/2 .