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AP Calculus AB Practice Quiz: Defining Continuity at a Point

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

According to the formal definition, a function f is continuous at a point x=c if three specific conditions are met. Which of the following is NOT one of these required conditions?

All Questions (7)

According to the formal definition, a function f is continuous at a point x=c if three specific conditions are met. Which of the following is NOT one of these required conditions?

A) f(c) exists.

B) The derivative of f exists at x=c.

C) lim (x->c) f(x) exists.

D) lim (x->c) f(x) = f(c).

Correct Answer: B

The definition of continuity at a point x=c requires that (1) f(c) exists, (2) the limit of f(x) as x approaches c exists, and (3) the limit equals the function's value at c. Differentiability at x=c is a stronger condition; a function can be continuous at a point without being differentiable there (e.g., a sharp corner).

Let f be the function defined by f(x) = (x^2 - 9) / (x - 3). Which of the following statements correctly justifies why f is not continuous at x=3?

A) lim (x->3) f(x) does not exist.

B) f(3) does not exist.

C) lim (x->3) f(x) is not a finite number.

D) lim (x->3) f(x) exists, but it is not equal to f(3).

Correct Answer: B

The first condition for continuity at a point c is that f(c) must exist. For the function f(x) = (x^2 - 9) / (x - 3), substituting x=3 results in a denominator of zero, making f(3) undefined. Since the first condition fails, the function is not continuous at x=3.

Consider the function g(x) defined as: g(x) = { x+2 if x < 1; 4 if x >= 1 }. Why is the function g not continuous at x=1?

A) g(1) does not exist.

B) lim (x->1) g(x) does not exist.

C) lim (x->1) g(x) exists, but it does not equal g(1).

D) The function g is continuous at x=1.

Correct Answer: B

To check for continuity at x=1, we evaluate the three conditions. First, g(1) = 4, so the function is defined. Second, we check if the limit exists by examining the one-sided limits. The left-hand limit is lim (x->1-) (x+2) = 3. The right-hand limit is lim (x->1+) (4) = 4. Since the left-hand and right-hand limits are not equal, the overall limit lim (x->1) g(x) does not exist. This violates the second condition for continuity.

A function h is defined such that lim (x->5) h(x) = 10 and h(5) = 2. Which statement provides a correct justification regarding the continuity of h at x=5?

A) The function h is continuous at x=5 because h(5) exists and the limit as x approaches 5 exists.

B) The function h is not continuous at x=5 because h(5) does not exist.

C) The function h is not continuous at x=5 because lim (x->5) h(x) does not exist.

D) The function h is not continuous at x=5 because lim (x->5) h(x) != h(5).

Correct Answer: D

The three conditions for continuity at x=c are: (1) f(c) exists, (2) lim (x->c) f(x) exists, and (3) lim (x->c) f(x) = f(c). Here, h(5)=2, so condition (1) is met. We are given lim (x->5) h(x) = 10, so condition (2) is met. However, since 10 != 2, the third condition, lim (x->5) h(x) = h(5), is not met. Therefore, the function is not continuous at x=5.

Let f be the function defined by: f(x) = { (x^2 - 16)/(x - 4) if x != 4; k if x = 4 }. For what value of the constant k is the function f continuous at x=4?

A) 0

B) 4

C) 8

D) 16

Correct Answer: C

For f to be continuous at x=4, the definition requires that lim (x->4) f(x) = f(4). First, we find the limit: lim (x->4) (x^2 - 16)/(x - 4) = lim (x->4) (x-4)(x+4)/(x-4) = lim (x->4) (x+4) = 4+4 = 8. The value of the function at x=4 is f(4)=k. To satisfy the definition of continuity, we must set the limit equal to the function value: k = 8.

The graph of a function f has a jump discontinuity at x=2. Based on the definition of continuity, which of the following statements must be the justification for this discontinuity?

A) f(2) is undefined.

B) The limit of f(x) as x approaches 2 does not exist.

C) The limit of f(x) as x approaches 2 exists, but it is not equal to f(2).

D) The function is not defined for x < 2.

Correct Answer: B

A jump discontinuity at x=c occurs when the left-hand limit and the right-hand limit both exist but are not equal. Because the one-sided limits are not equal, the overall limit, lim (x->c) f(x), does not exist. This violates the second condition of the definition of continuity.

Let f be a function for which it is known that f(c) = L. Which of the following statements provides a sufficient justification to conclude that f is continuous at x=c?

A) The limit of f(x) as x approaches c from the left is L.

B) The limit of f(x) as x approaches c from the right is L.

C) The limit of f(x) as x approaches c exists.

D) The limit of f(x) as x approaches c is L.

Correct Answer: D

The definition of continuity at x=c requires three conditions to be met. The premise f(c) = L establishes that condition (1), f(c) exists, is met. The statement in option D, lim (x->c) f(x) = L, establishes that condition (2), the limit exists, is met. It also directly connects the limit to the function value, satisfying condition (3), lim (x->c) f(x) = f(c), because both are equal to L. Therefore, this single statement is sufficient to justify continuity.