If a function f has a jump discontinuity at x = a, what can be said about lim_{x->a} f(x)?

If a function f has a jump discontinuity at x = a, what can be said about lim_{x->a} f(x)?

• A. The limit exists and equals f(a).
• B. The limit exists but may not equal f(a).
• C. The limit does not exist.
• D. The limit equals the average of left and right limits.

Answer: (C)

A jump discontinuity happens when approaching the point from the left and from the right leads to finite values that are different. In that case, the left-hand limit and the right-hand limit both exist, but they do not agree. Since a two-sided limit requires these one-sided limits to match, the limit as x approaches a does not exist. The actual value of the function at a, f(a), doesn’t fix the issue—the function can jump regardless of what f(a) is. So the correct takeaway is that the two-sided limit fails to exist at a jump discontinuity.

A jump discontinuity happens when approaching the point from the left and from the right leads to finite values that are different. In that case, the left-hand limit and the right-hand limit both exist, but they do not agree. Since a two-sided limit requires these one-sided limits to match, the limit as x approaches a does not exist. The actual value of the function at a, f(a), doesn’t fix the issue—the function can jump regardless of what f(a) is. So the correct takeaway is that the two-sided limit fails to exist at a jump discontinuity.