Which statement correctly characterizes a jump discontinuity at x = a?

Which statement correctly characterizes a jump discontinuity at x = a?

• A. The left-hand limit equals the right-hand limit.
• B. The function is continuous at a.
• C. The left-hand and right-hand limits both exist but are not equal.
• D. The limit equals the average of left and right limits.

Answer: (C)

A jump discontinuity occurs when approaching the point from the left and from the right you get two finite limits, but those two limits are not the same. That makes the function jump from one level to another at x = a, so there isn’t a single limit there and the function isn’t continuous at a. The value of the function at a may be anything, and continuity would require this value to equal the common (shared) value of the one-sided limits, which is not the case here. If the left- and right-hand limits were equal, you wouldn’t have a jump; you’d either have continuity (if f(a) matches that common value) or a removable discontinuity (if f(a) differs). The average of the two one-sided limits isn’t relevant to characterizing a jump.

A jump discontinuity occurs when approaching the point from the left and from the right you get two finite limits, but those two limits are not the same. That makes the function jump from one level to another at x = a, so there isn’t a single limit there and the function isn’t continuous at a. The value of the function at a may be anything, and continuity would require this value to equal the common (shared) value of the one-sided limits, which is not the case here.

If the left- and right-hand limits were equal, you wouldn’t have a jump; you’d either have continuity (if f(a) matches that common value) or a removable discontinuity (if f(a) differs). The average of the two one-sided limits isn’t relevant to characterizing a jump.