scholarly journals A Formalization of Properties of Continuous Functions on Closed Intervals

2020 ◽  
pp. 272-280
Author(s):  
Yaoshun Fu ◽  
Wensheng Yu
Keyword(s):  
Continuous Functions ◽  
Closed Intervals

DEFINITION AND CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750048 ◽  
Author(s):  
Y. S. LIANG
Keyword(s):  
Fractal Dimension ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Nondifferentiable Functions ◽  
Differentiable Functions ◽  
Closed Intervals ◽  
Bounded Variation Functions

The present paper mainly investigates the definition and classification of one-dimensional continuous functions on closed intervals. Continuous functions can be classified as differentiable functions and nondifferentiable functions. All differentiable functions are of bounded variation. Nondifferentiable functions are composed of bounded variation functions and unbounded variation functions. Fractal dimension of all bounded variation continuous functions is 1. One-dimensional unbounded variation continuous functions may have finite unbounded variation points or infinite unbounded variation points. Number of unbounded variation points of one-dimensional unbounded variation continuous functions maybe infinite and countable or uncountable. Certain examples of different one-dimensional continuous functions have been given in this paper. Thus, one-dimensional continuous functions are composed of differentiable functions, nondifferentiable continuous functions of bounded variation, continuous functions with finite unbounded variation points, continuous functions with infinite but countable unbounded variation points and continuous functions with uncountable unbounded variation points. In the end of the paper, we give an example of one-dimensional continuous function which is of unbounded variation everywhere.

scholarly journals Approximation by modified Jain–Baskakov operators

2020 ◽  
Vol 27 (3) ◽  
pp. 403-412
Author(s):  
Vishnu Narayan Mishra ◽  
Preeti Sharma ◽  
Marius Mihai Birou
Keyword(s):  
Weighted Approximation ◽  
Approximation Order ◽  
Continuous Functions ◽  
Basis Functions ◽  
Test Functions ◽  
Approximation Properties ◽  
Baskakov Operators ◽  
Closed Intervals ◽  
Direct Results ◽  
Modified Forms

AbstractIn the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions {e_{0}} and {e_{1}}. It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

Classical Analysis

2019 ◽  
pp. 51-69
Author(s):  
John Stillwell
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Uniform Continuity ◽  
Cantor Set ◽  
Binary Trees ◽  
Real Numbers ◽  
Limit Concept ◽  
Closed Intervals ◽  
Basic Concepts

This chapter explores the basic concepts that arise when real numbers and continuous functions are studied, particularly the limit concept and its use in proving properties of continuous functions. It gives proofs of the Bolzano–Weierstrass and Heine–Borel theorems, and the intermediate and extreme value theorems for continuous functions. Also, the chapter uses the Heine–Borel theorem to prove uniform continuity of continuous functions on closed intervals, and its consequence that any continuous function is Riemann integrable on closed intervals. In several of these proofs there is a construction by “infinite bisection,” which can be recast as an argument about binary trees. Here, the chapter uses the role of trees to construct an object—the so-called Cantor set.

Discontinuities of the pressure for piecewise monotonic interval maps

2001 ◽  
Vol 21 (1) ◽  
pp. 197-232 ◽  
Author(s):  
PETER RAITH
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Finite Union ◽  
Topological Pressure ◽  
Stability Conditions ◽  
Small Perturbations ◽  
Interval Maps ◽  
Closed Intervals ◽  
Piecewise Monotonic ◽  
Special Case

For a piecewise monotonic map T:X\to{\Bbb R}, where X is a finite union of closed intervals, define R(T)= \bigcap_{n=0}^{\infty}\overline{T^{-n}X}. The influence of small perturbations of T on the dynamical system (R(T),T) is investigated. If P is a finite and T-invariant subset of R(T), and if f_0:P\to{\Bbb R} is a non-negative continuous function, then it is proved that the infimum of the topological pressure p(R(T),T,f) over all non-negative continuous functions f:X\to{\Bbb R} with f|_P=f_0 equals the maximum of h_{\text{\rm top}}(R(T),T) and p(P,T,f_0). This result is used to obtain stability conditions, which are equivalent to the upper semi-continuity of the topological pressure for every continuous function f:X\to{\Bbb R}. In the case of a continuous piecewise monotonic map T:X\to{\Bbb R} one of these stability conditions is: there exists no endpoint of an interval of monotonicity of T which is periodic and contained in the interior of X. Furthermore, these results are applied to monotonic mod one transformations, another special case of piecewise monotonic maps.

FRACTAL DIMENSION OF CERTAIN CONTINUOUS FUNCTIONS OF UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Y. S. LIANG ◽  
W. Y. SU
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Bounded Variation ◽  
Closed Interval ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
One Dimensional ◽  
Closed Intervals ◽  
Bounded Variation Functions

Continuous functions on closed intervals are composed of bounded variation functions and unbounded variation functions. Fractal dimension of continuous functions with bounded variation must be one-dimensional (1D). While fractal dimension of continuous functions with unbounded variation may be 1 or not. Certain continuous functions of unbounded variation whose fractal dimensions are 1 have been mainly investigated in the paper. A continuous function on a closed interval with finite unbounded variation points has been proved to be 1D. Furthermore, we deal with continuous functions which have infinite unbounded variation points and part of them have been proved to be 1D. Certain examples of 1D continuous functions which have uncountable unbounded variation points have been given in the present paper.

scholarly journals I-approximately continuous functions

2015 ◽  
Vol 62 (1) ◽  
pp. 45-55
Author(s):  
Jacek Hejduk ◽  
Anna Loranty ◽  
Renata Wiertelak
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Density Topology ◽  
Closed Intervals

Abstract In this paper, density-like points and density-like topology connected with a sequence I = {In}n∊ℕ of closed intervals tending to 0 will be considered. We introduce the notion of an I -approximately continuous function associated with this kind of density points. Moreover, we present some properties of these functions and we demonstrate their connection with continuous functions with respect to this kind of density topology.

On some properties of 𝓙-approximately continuous functions

2017 ◽  
Vol 67 (6) ◽  
Author(s):  
Jacek Hejduk ◽  
Renata Wiertelak
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Density Topology ◽  
Closed Intervals

AbstractIn this paper we will consider 𝓙-density topology connected with a sequence 𝓙 of closed intervals tending to 0 and a 𝓙-approximately continuous function associated with that kind of density points. It will be the continuation of the investigations started in “𝓙-

scholarly journals On common fixed points, periodic points, and recurrent points of continuous functions

2003 ◽  
Vol 2003 (39) ◽  
pp. 2465-2473 ◽  
Author(s):  
Aliasghar Alikhani-Koopaei
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Dense Subset ◽  
Periodic Points ◽  
Continuous Functions ◽  
Common Fixed Points ◽  
Closed Intervals ◽  
Disjoint Sets ◽  
The Common ◽  
Class Of Functions

It is known that two commuting continuous functions on an interval need not have a common fixed point. However, it is not known if such two functions have a common periodic point. we had conjectured that two commuting continuous functions on an interval will typically have disjoint sets of periodic points. In this paper, we first prove thatSis a nowhere dense subset of[0,1]if and only if{f∈C([0,1]):Fm(f)∩S¯≠∅}is a nowhere dense subset ofC([0,1]). We also give some results about the common fixed, periodic, and recurrent points of functions. We consider the class of functionsfwith continuousωfstudied by Bruckner and Ceder and show that the set of recurrent points of such functions are closed intervals.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

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