Abstract Let I be a compact interval and f : I ⟶ I be continuous. Assume that F ( I ) is the set of fuzzy numbers on I, and that f ^ : F ( I ) ⟶ F ( I ) is the Zadeh′s extension of f, and that lim ← { F ( I ) , f ^ } is the inverse limit space of ( F ( I ) , f ^ ) , and σ f ^ : lim ← { F ( I ) , f ^ } ⟶ lim ← { F ( I ) , f ^ } is the left shift map. In this paper, we study the pointwise equicontinuity of f ^ and show that the following statements are equivalent: (1) f ^ is pointwise equicontinuous. (2) For some infinite subsequence S of positive integers, f ^ is S pointwise equicontinuous. (3) { f ^ 2 n } n = 1 ∞ is uniformly convergent on F ( I ) . (4) σ f ^ is a periodic map with period 2.