The fixed-point set and the convergence set of a continuous function

Abstract

Let X be a Hausdorff space and f:X --> X a continuous function. The fixed-point set of f, denoted by F(f), is the set of all points x E X such that f(x) = x. The convergence set of f, denoted by C(f), is the set of x E X such that the sequence (fn(x)) converges in X where fn = f o f o ... o f {n times}. We define f infinity : C(f) --> F(f) by f infinity (x) = lim(n-->infinity) fn(x) for all x E C(f). In this thesis, we study basic properties of F(f) and C(f) and show that f infinity is continuous if and only if f infinity is continuous at each point in F(f). Moreover, we prove that if X is a metric space and f is quasi-nonexpansive then f infinity is continuous.