Δ-related functions and generalized inverse limits

For any continuous single-valued functions f, g : [0, 1] → [0, 1] we define upper semicontinuous set-valued functions F,G : [0, 1] ⊸ [0, 1] by their graphs as the unions of the diagonal ∆ and the graphs of setvalued inverses of f and g respectively. We introduce when two functions are ∆-related and show that if f and g are ∆-related, then the inverse limits lim − ⊸ F and lim − ⊸ G are homeomorphic. We also give conditions under which lim − ⊸ G is a quotient space of lim − ⊸ F .