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 Decomposition of Topologies Which Characterize the Upper and Lower Semicontinuous Limits of Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Agata Caserta
Keyword(s):  
Continuous Function ◽  
Lower Semicontinuity ◽  
Pointwise Convergence ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Point Of View ◽  
Limit Function ◽  
Statistical Point ◽  
Upper And Lower Semicontinuity

We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that (from the statistical point of view) there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology.

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-

ON THE DECOMPOSITION OF CONTINUOUS FUNCTIONS AND DIMENSIONS

Fractals ◽  
2019 ◽  
Vol 28 (01) ◽  
pp. 2050007
Author(s):  
JIA LIU ◽  
DEZHI LIU
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Present Author ◽  
Continuous Functions ◽  
Box Dimension ◽  
Real Numbers ◽  
Dimension Formula

In this paper, we consider decomposition of continuous functions in [Formula: see text] in terms of Hausdorff dimension and lower box dimension. Precisely, we show that, given real numbers [Formula: see text], any real-valued continuous function in [Formula: see text] can be decomposed into a sum of two real-valued continuous functions each having a graph of Hausdorff dimension [Formula: see text] and lower box dimension [Formula: see text]. This generalizes a theorem of Wingren, also Wu and the present author. We also consider the arbitrary decomposition of continuous functions in terms of Hausdorff dimension and lower box dimension.

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.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

scholarly journals A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽  
2021 ◽  
Author(s):  
Péter Vrana
Keyword(s):  
Compact Hausdorff Space ◽  
Polynomial Growth ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Equivalent Condition ◽  
Real Numbers ◽  
Asymptotic Spectrum ◽  
Ring Of Continuous Functions ◽  
Archimedean Property ◽  
Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

Cell cycle controls and the role of nerves and the regenerate epithelium in urodele forelimb regeneration: possible modifications of basic concepts

1987 ◽  
Vol 65 (8) ◽  
pp. 739-749 ◽  
Author(s):  
Roy A. Tassava ◽  
David J. Goldhamer ◽  
Bruce L. Tomlinson
Keyword(s):  
Cell Cycle ◽  
Monoclonal Antibody ◽  
Limb Regeneration ◽  
Blastema Formation ◽  
Basic Concepts ◽  
Regenerate Epithelium ◽  
Early Wound ◽  
Skin Epidermis ◽  
Wound Epithelium

Data from pulse and continuous labeling with [3H]thymidine and from studies with monoclonal antibody WE3 have led to the modification of existing models and established concepts pertinent to understanding limb regeneration. Not all cells of the adult newt blastema are randomly distributed and actively progressing through the cell cycle. Instead, many cells are in a position that we have designated transient quiescence (TQ) and are not actively cycling. We postulate that cells regularly leave the TQ population and enter the actively cycling population and vice versa. The size of the TQ population may be at least partly determined by the quantity of limb innervation. Larval Ambystoma may have only a small or nonexisting TQ, thus accounting for their rapid rate of regeneration. Examination of reactivity of monoclonal antibody WE3 suggests that the early wound epithelium, which is derived from skin epidermis, is later replaced by cells from skin glands concomitant with blastema formation. WE3 provides a useful tool to further investigate the regenerate epithelium.

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