  We discuss the problem of when a continuous map between topological spaces
induces a continuous function between their respective hyperspaces. We
characterize the continuity of the induced function in the case of the Fell and
Attouch-Wets hyperspaces. Additionally we explore the problem of whether a
continuous action of a topological group $G$ on a topological space $X$ induces
a continuous action on $CL(X)$.
