


Paper Title:
Uniform Continuity and kConvexity
Author(s):
Adel Afif Abdelkarim
Mathematics Department, Faculty of Science,
J
erash University, Jerash
Jordan.
Abstract:
A closed arcwiseconnected subset A of R^{n} is called kconvex if for each positive number a and for all elements x and y in A there is a positive number b such that if the norm of xy is less than or equal to b then the length of the shortest curve l(x,y) in A is less than k times the norm of xy plus a. We show that a union of two non disjoint closed finite convex subsets need not be kconvex. Let f(x) be a uniformly continuous functions on a finite number of closed subsets A_{1},...,A_{n} of R^{n} such that the union of A_{j},...,A_{n},j=1,...,n1 is kconvex. We show that f is uniformly continuous on the union of the sets A_{i},i=1,...,n. We give counter examples if this condition is not satisfied. As a corollary we show that if f(x) is uniformly continuous on each of two closed convex sets A,B then f(x) is uniformly continuous on the union of A and B.
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