Connectedness of inverse limits with functions f where either f or fi−1 is a union of continuum-valued functions

We establish a general theorem for connectedness of the inverse limit X of an inverse sequence { X i , f i } i ≥ 1 on metric continua with surjective upper semi-continuous set-valued bonding functions, where for each i ≥ 1 , f i has a connected graph, and either f i or f i − 1 is a union of continuum-valued functions. Properties of certain set-valued functions from the factor spaces onto partial graphs in the inverse sequence imply connectedness of X.