scholarly journals A method of joining piecewise functions to produce continuous functions of difficult data

2021 ◽  
Author(s):  
Stefan T Orszulik
Keyword(s):  
Continuous Function ◽  
Original Data ◽  
Continuous Functions ◽  
Dummy Variables ◽  
Modelling Data ◽  
Piecewise Functions

Abstract This article describes a method of modelling data that involves splitting the curve into two (or more) and creating separate piecewise functions for each part; these functions are then concatenated via a linking function to create one overall continuous function that better describes the original data than is otherwise achievable. The linking function is able to do this by separating the original two (or more) subfunctions so that they are each active in only the relevant portion of the overall curve without the use of dummy variables. The final result is a continuous function in which it is straightforward to smooth the transition at the knot between the piecewise subfunctions. In addition, the piecewise subfunctions do not need to align at the knot since the degree of smoothing is very readily controlled. All types of functions may be concatenated so that the method is flexible and relatively simple to apply.

scholarly journals A Method of Joining Piecewise Functions to Produce Continuous Functions of Difficult Data

2021 ◽  
Author(s):  
Stefan Orszulik
Keyword(s):  
Continuous Function ◽  
Original Data ◽  
Continuous Functions ◽  
Dummy Variables ◽  
Modelling Data ◽  
Piecewise Functions

This article describes a method of modelling data that involves splitting the curve into two (or more) and creating separate piecewise functions for each part; these functions are then concatenated via a linking function to create one overall continuous function that better describes the original data than is otherwise achievable. The linking function is able to do this by separating the original two (or more) subfunctions so that they are each active in only the relevant portion of the overall curve without the use of dummy variables. The final result is a continuous function in which it is straightforward to smooth the transition at the knot between the piecewise subfunctions. In addition, the piecewise subfunctions do not need to align at the knot since the degree of smoothing is readily controlled. All types of functions may be concatenated so that the method is flexible and relatively simple to apply.

scholarly journals A Method of Joining Piecewise Functions to Produce Continuous Functions of Difficult Data

2021 ◽  
Author(s):  
Stefan Orszulik
Keyword(s):  
Continuous Function ◽  
Original Data ◽  
Continuous Functions ◽  
Dummy Variables ◽  
Modelling Data ◽  
Piecewise Functions

This article describes a method of modelling data that involves splitting the curve into two (or more) and creating separate piecewise functions for each part; these functions are then concatenated via a linking function to create one overall continuous function that better describes the original data than is otherwise achievable. The linking function is able to do this by separating the original two (or more) subfunctions so that they are each active in only the relevant portion of the overall curve without the use of dummy variables. The final result is a continuous function in which it is straightforward to smooth the transition at the knot between the piecewise subfunctions. In addition, the piecewise subfunctions do not need to align at the knot since the degree of smoothing is readily controlled. All types of functions may be concatenated so that the method is flexible and relatively simple to apply.

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 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.

A Regular Space on Which Every Real-Valued Function with a Closed Graph is Constant

1989 ◽  
Vol 32 (4) ◽  
pp. 417-424 ◽  
Author(s):  
Ivan Baggs
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Regular Space ◽  
Closed Graph

AbstractAn example is given of a regular space on which every real-valued function with a closed graph is constant. It was previously known that there are regular spaces on which every continuous function is constant. It is also shown here that there are regular spaces that support only constant real-valued continuous functions, but support non-constant real-valued functions with a closed graph.

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

scholarly journals On Weighted Polynomial Approximation With Gaps

2005 ◽  
Vol 178 ◽  
pp. 55-61 ◽  
Author(s):  
Guantie Deng
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Polynomial Approximation ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Weighted Polynomial Approximation ◽  
Necessary And Sufficient Condition ◽  
Weighted Banach Space ◽  
Nonnegative Continuous Function ◽  
Necessary And Sufficient

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

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