The Analysis of Quasi-Continuous Functions
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We give examples of quasi-continuous functions whose second iterates are discontinuous everywhere in the domain. A function is quasi-continuous if the inverse image of every open set in the codomain is quasi-open in the domain. A set is quasi-open if it is a subset of the interior of that set. Crannell, Frantz and LeMasurier (CFL)  showed that if a function ƒ : X -> Y is quasi-continuous and semi-open, then the second iterate of ƒ has a dense set of continuity points in X. A function is said to be semi-open if the image of every non-empty, open subset of the domain contains a non-empty, open subset of the range. We show that the requirement of being semi-open is necessary by giving two examples of quasi-continuous functions, with discontinuous second iterates, that are not quopen. A function that is both quasi-continuous and semi-open is said to be quopen. We study these functions in the hope of extending known results for chaotic dynamical systems.
Franklin and Marshall College Archives, Undergraduate Honors Thesis 2007
- F&M Theses Collection