Continuity of the maps f↦∪x∈Iω(x, f) and f↦{ω(x, f): x∈I}

We study the behavior of two maps in an effort to better understand the stability of ωlimit sets ω(x, f ) as we perturb either x or f , or both. The first map is the set-valued function Λ taking f in C(I ,I) to its collection of ω-limit points Λ( f ) = ⋃x∈I ω(x, f ), and the second is the map Ω taking f in C(I ,I) to its collection of ω-limit sets Ω( f )= {ω(x, f ) : x ∈ I}.We characterize those functions f in C(I ,I) at which each of our maps Λ and Ω is continuous, and then go on to show that both Λ and Ω are continuous on a residual subset of C(I ,I). We then investigate the relationship between the continuity of Λ and Ω at some function f in C(I ,I) with the chaotic nature of that function.