Calculus: Point Continuity vs. Interval Continuity
I am confused about continuity at a point versus continuity of endpoints that define a closed interval. While taking the Unit Test for Limits and Continuity, I was given two questions, one of which involved continuity of a point and one that involved continuity over a closed interval. The given answers seem contradictory to me. See the included screen shots for each question below. One question asked whether g(x) is continuous at the point x = 0. The correct answer was no, as the left and right one sided limits did not yield the same value. The second question asked over what intervals g(x) is continuous. The correct answer was the closed interval [-6, -1]. The closed interval contained a jump discontinuity at -1. It seems contradictory to me that the closed interval would be considered continuous, yet the point (1,-1) of that same interval would not be considered continuous. It seems as if the criteria for interval continuity is less strict than that for point continuity. That is, the end points of a closed interval only need the respective one sided limit to exist, where as the continuity of a point requires both one sided limits to exist. Consequently, although a closed interval is considered continuous, the points at each end of that closed interval may not be continuous. Am I understanding continuity criteria correctly?
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