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-

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.

Almost Somewhat Near Continuity and Near Regularity

2021 ◽  
Vol 7 (1) ◽  
pp. 88-99
Author(s):  
Zanyar A. Ameen
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
One To One ◽  
Nearly Continuous

AbstractThe notions of almost somewhat near continuity of functions and near regularity of spaces are introduced. Some properties of almost somewhat nearly continuous functions and their connections are studied. At the end, it is shown that a one-to-one almost somewhat nearly continuous function f from a space X onto a space Y is somewhat nearly continuous if and only if the range of f is nearly regular.

scholarly journals A note on weakly θ-continuous functions

1989 ◽  
Vol 12 (1) ◽  
pp. 9-13 ◽  
Author(s):  
M. Mrševic ◽  
I. L. Reilly
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
New Class ◽  
Weakly Continuous ◽  
Class Of Functions

Recently a new class of functions between topological spaces, called weaklyθ-continuous functions, has been introduced and studied. In this paper we show how an appropriate change of topology on the domain of a weaklyθ-continuous function reduces it to a weakly continuous function. This paper examines some of the consequences of this result.

scholarly journals Density topology and pointwise convergence

2003 ◽  
Vol 4 (2) ◽  
pp. 509 ◽  
Author(s):  
Wladyslaw Wilczynski
Keyword(s):  
Pointwise Convergence ◽  
Continuous Functions ◽  
Density Topology ◽  
Topology Of Pointwise Convergence

<p>We shall show that the space of all approximately continuous functions with the topology of pointwise convergence is not homeomorphic to its category analogue.</p>

scholarly journals Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

scholarly journals ‎INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO ‎‎COMPARABLE CONTRA-$\alpha-$CONTINUOUS (CONTRA-$C-$CONTINUOUS) FUNCTIONS

2019 ◽  
pp. 13
Author(s):  
Majid Mirmiran ◽  
Binesh Naderi
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

‎A necessary and sufficient condition in terms of lower cut sets ‎are given for the insertion of a contra-continuous function ‎between two comparable real-valued functions on such topological ‎spaces that kernel of sets are open‎. 

scholarly journals On the PropertyN-1

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Stanisław Kowalczyk ◽  
Małgorzata Turowska
Keyword(s):  
Continuous Function ◽  
Lebesgue Measure ◽  
Continuous Functions ◽  
Approximate Derivative ◽  
Banach's Theorem ◽  
Full Lebesgue Measure

We construct a continuous functionf:[0,1]→Rsuch thatfpossessesN-1-property, butfdoes not have approximate derivative on a set of full Lebesgue measure. This shows that Banach’s Theorem concerning differentiability of continuous functions with Lusin’s property(N)does not hold forN-1-property. Some relevant properties are presented.

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