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Let X be a continuum. Let CðX Þ be the hyperspace of all closed, connected and nonempty subsets of X , with the Hausdor¤ metric. For a mapping f : X ! Y between continua, let Cð f Þ : CðX Þ ! CðY Þ be the induced mapping by f , given by Cð f ÞðAÞ ¼ f ðAÞ. In this paper we study the hyperspace CðX Þ ¼ fCðAÞ : A A CðX Þg as a subspace of CðCðX ÞÞ, and define an induced function Cð f Þ between CðX Þ and CðY Þ. We prove some relationships between the functions f , Cð f Þ and Cð f Þ for the following classes of mapping: confluent, light, monotone and weakly confluent.