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AP Calculus BC 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, which of the following conditions must be met for a function f to be continuous at a point x=c?

All Questions (7)

According to the formal definition, which of the following conditions must be met for a function f to be continuous at a point x=c?

A) Only that f(c) exists.

B) Only that the limit of f(x) as x approaches c exists.

C) Only that the limit of f(x) as x approaches c is equal to f(c).

D) f(c) must exist, the limit of f(x) as x approaches c must exist, and the limit must equal f(c).

Correct Answer: D

The definition of continuity at a point requires all three conditions to be satisfied: (1) f(c) exists, (2) the limit of f(x) as x approaches c exists, and (3) the limit is equal to the function's value at c.

The function g(x) = (x^2 - 9) / (x - 3) is not continuous at x=3. Which part of the three-part definition of continuity at a point fails?

A) g(3) does not exist.

B) The limit of g(x) as x approaches 3 does not exist.

C) The limit of g(x) as x approaches 3 exists, but it is not equal to g(3).

D) All three parts of the definition fail.

Correct Answer: A

For the function g(x) at x=3, substituting 3 into the expression results in a denominator of zero. Therefore, g(3) is undefined. Since the first condition for continuity, that f(c) must exist, is not met, the function is not continuous at x=3.

Consider a function h(x) for which lim (x->2-) h(x) = 4 and lim (x->2+) h(x) = 6. Based on this information, which justification explains why h(x) cannot be continuous at x=2?

A) h(2) is not defined.

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

C) The limit of h(x) as x approaches 2 is not equal to h(2).

D) The function is increasing too rapidly at x=2.

Correct Answer: B

For the second condition of continuity, the limit of h(x) as x approaches 2 must exist. This requires the left-hand limit to equal the right-hand limit. Since the left-hand limit (4) is not equal to the right-hand limit (6), the overall limit does not exist, and the function is not continuous at x=2.

Let f be a function for which f(5) = 10 and the limit of f(x) as x approaches 5 is 8. Which statement correctly justifies why f is not continuous at x=5?

A) f(5) does not exist.

B) The limit of f(x) as x approaches 5 does not exist.

C) The limit of f(x) as x approaches 5 is not equal to f(5).

D) Both f(5) and the limit of f(x) as x approaches 5 do not exist.

Correct Answer: C

In this case, the first two conditions for continuity are met: f(5) exists (it is 10) and the limit of f(x) as x approaches 5 exists (it is 8). However, the third condition, that the limit must equal the function value, fails because 8 ≠ 10. This type of discontinuity is a removable discontinuity.

If a function g is known to be continuous at x=c, which of the following statements provides the most complete and accurate conclusion based on the definition of continuity?

A) The graph of g must be a smooth curve at x=c.

B) g(c) must be defined, but the limit as x approaches c may not exist.

C) The limit of g(x) as x approaches c must exist, but it does not have to equal g(c).

D) The value that g(x) approaches as x gets arbitrarily close to c is precisely g(c).

Correct Answer: D

This statement is a conceptual rephrasing of the third and final condition for continuity, lim (x->c) g(x) = g(c). For a function to be continuous, this condition must be met, which implies that the first two conditions (g(c) exists and the limit exists) are also satisfied. It is the most complete description of the consequence of continuity.

Let f be the function defined by f(x) = { x^2 + 3, if x ≠ 1; 5, if x = 1 }. Why is f not continuous at x=1?

A) f(1) is not defined.

B) The limit of f(x) as x approaches 1 does not exist.

C) The limit of f(x) as x approaches 1 exists, but it is not equal to f(1).

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

Correct Answer: C

We must test the three conditions for continuity at x=1. (1) f(1) exists and is defined as 5. (2) The limit of f(x) as x approaches 1 is found using the expression x^2 + 3, which gives lim (x->1) (x^2 + 3) = 1^2 + 3 = 4. The limit exists. (3) We compare the limit and the function value: the limit is 4 and f(1) is 5. Since 4 ≠ 5, the third condition for continuity fails.

A student justifies that a function f is continuous at x=a because the limit of f(x) as x approaches a exists. Is this justification sufficient?

A) Yes, if the limit exists, the function must be continuous.

B) No, because f(a) must also exist.

C) No, because f(a) must exist and the limit must be equal to f(a).

D) Yes, but only if the limit is a non-zero number.

Correct Answer: C

The existence of the limit is only one of the three required conditions for continuity. For f to be continuous at x=a, f(a) must also be defined, and critically, the value of the limit must be exactly equal to the value of f(a). The student's justification is incomplete.