Let f denote a continuous map of a compact interval I to itself. A point x∈I is called a γ-limit point of f if it is both an ω-limit point and a α-limit point of some point y∈I. Let I’denote the set of γ-limit points. In the present paper, we show that (1) P-I’ is either empty or countably infinite, where P denotes the closure of the set P of periodic points, (2) x∈I is a γ-limit point if and only if there exist y1 and y2 in I such that x is an ω-limit points of y1, and y1 is an ω-limit point of y2, and if and only if there exists a sequence y1, y2, … of points in I such that x is an ω-limit point of y1, and y1 is an ω-limit point of yi+1 for every i≥1, and (3) the period of each periodic point of f is a power of 2 if and only if every γ-limit point is recurrent.