The Analysis of QuasiContinuous Functions

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MathematicsAbstract
We give examples of quasicontinuous functions whose second iterates are discontinuous everywhere in the domain. A function is quasicontinuous if the inverse image of every open set in the codomain is quasiopen in the domain. A set is quasiopen if it is a subset of the interior of that set. Crannell, Frantz and LeMasurier (CFL) [1] showed that if a function ƒ : X > Y is quasicontinuous and semiopen, then the second iterate of ƒ has a dense set of continuity points in X. A function is said to be semiopen if the image of every nonempty, open subset of the domain contains a nonempty, open subset of the range.
We show that the requirement of being semiopen is necessary by giving two examples of quasicontinuous functions, with discontinuous second iterates, that are not quopen. A function that is both quasicontinuous and semiopen is said to be quopen. We study these functions in the hope of extending known results for chaotic dynamical systems.
Description
Franklin and Marshall College Archives, Undergraduate Honors Thesis 2007
Collections
 F&M Theses Collection [279]
Date
2007Author
Alam, Muhammad S.
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Show full item recordAuthor  Alam, Muhammad S.  
Date Accessioned  20080228T17:04:09Z  
Date Available  20080228T17:04:09Z  
Date of Issue  2007  
Identifier (URI)  http://hdl.handle.net/11016/4192  
Description  Franklin and Marshall College Archives, Undergraduate Honors Thesis 2007  en 
Description  We give examples of quasicontinuous functions whose second iterates are discontinuous everywhere in the domain. A function is quasicontinuous if the inverse image of every open set in the codomain is quasiopen in the domain. A set is quasiopen if it is a subset of the interior of that set. Crannell, Frantz and LeMasurier (CFL) [1] showed that if a function ƒ : X > Y is quasicontinuous and semiopen, then the second iterate of ƒ has a dense set of continuity points in X. A function is said to be semiopen if the image of every nonempty, open subset of the domain contains a nonempty, open subset of the range. We show that the requirement of being semiopen is necessary by giving two examples of quasicontinuous functions, with discontinuous second iterates, that are not quopen. A function that is both quasicontinuous and semiopen is said to be quopen. We study these functions in the hope of extending known results for chaotic dynamical systems.  en 
Sponsorship  Franklin and Marshall College, Mathematics Department  en 
Language  en_US  en 
Subject  Mathematics  
Title  The Analysis of QuasiContinuous Functions  en 
Type  Thesis  en 