  In this paper, we consider inverse limits of $[0,1]$ using upper
semicontinuous set-valued functions. We introduce two generalizations of the
Intermediate Value Property and prove that inverse limits with upper
semicontinuous set-valued bonding functions are connected if the bonding
functions are surjective, have connected graphs, and have either generalization
of the Intermediate Value Property. Examples are given to demonstrate that if
any of the conditions is dropped, the result does not hold in general. An
example is given to show that an inverse limit may be connected even if the
bonding functions do not have either Intermediate Value Property. Further, we
compare the structures of set-valued functions with the two types of the
Intermediate Value Property.
