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Department Colloquium – Dr. James Foran, University of Missouri Kansas City – “Constructing Continuous Functions”

The Department of Mathematics & Statistics would like to welcome:

Dr. James Foran on Thursday, March 7th at 11:30am in SCP 229.

Title: Constructing Continuous Functions

Abstract:  This talk will be concerned with continuous real valued functions defined on the unit interval.  After presenting several statements equivalent to continuity, the fact that the graph must be compact, that is, closed and bounded, is used to produce geometrical constructions involving the intersection of sets that result in continuous functions.  Various properties of such functions will be considered and the graphs will be visually evident along with the properties claimed for the functions.  Definitions needed for the various properties will be provided before each graph is delineated.  Such properties include: Lipschitz but not differentiable on an uncountable set of points, the inverse image of each point is countable, nowhere differentiable with a graph that is the countable union of sets of finite length, non-sigma porous graph.  The coordinate functions for the Peano curve will also be presented with the geometrical means for constructing them.

James Foran is a professor emeritus living in Kansas City.  He received his B.A. at Marquette University in 1965, M.A. in 1967 and Ph.D. in 1973 from the University of Wisconsin-Milwaukee.  He was a professor at the University of Missouri-Kansas City from then until 2003.  He is the author of 45 research papers and a book entitled “Fundamentals of Real Analysis”, 476 pp. Marcel Dekker 1990.  He was editor of “The Real Analysis Exchange” from its inception in 1976 until 2002.  He was thesis advisor for six students who received their Ph.D from the University of Missouri-Kansas City.

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