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AP Calculus BC Practice Quiz: Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

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

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

Which of the following statements correctly describes the fundamental relationship between differentiability and continuity at a point?

All Questions (9)

Which of the following statements correctly describes the fundamental relationship between differentiability and continuity at a point?

A) If a function is differentiable at a point, then it must be continuous at that point.

B) If a function is continuous at a point, then it must be differentiable at that point.

C) A function is differentiable at a point if and only if it is continuous at that point.

D) Differentiability and continuity are independent properties of a function.

Correct Answer: A

According to the provided content, if a function is differentiable at a point, it is a guaranteed consequence that the function is also continuous at that point. The reverse is not always true.

A function f(x) is known to be continuous at x = a. What can be concluded about the differentiability of f(x) at x = a?

A) The function must be differentiable at x = a.

B) The function cannot be differentiable at x = a.

C) The function may or may not be differentiable at x = a.

D) The function must have a derivative of zero at x = a.

Correct Answer: C

The provided content states that a continuous function may fail to be differentiable. This means that while a continuous function can be differentiable, it is not a requirement. Therefore, no definitive conclusion can be drawn about its differentiability without more information.

Given that a function g(x) has a well-defined derivative g'(3), which of the following must be true?

A) g(x) is continuous at x = 3.

B) g(x) has a sharp corner at x = 3.

C) g(x) is not continuous at x = 3.

D) The continuity of g(x) at x = 3 cannot be determined.

Correct Answer: A

The existence of a derivative at a point means the function is differentiable at that point. A core principle is that if a function is differentiable at a point, it must also be continuous at that same point.

Which of the following statements is a logical consequence of the fact that differentiability implies continuity?

A) If a function is not differentiable at a point, it cannot be continuous at that point.

B) If a function is not continuous at a point, it cannot be differentiable at that point.

C) If a function is continuous at a point, it must be differentiable at that point.

D) A function is continuous only if it is differentiable.

Correct Answer: B

This question tests the contrapositive of the rule 'If a function is differentiable, then it is continuous.' The contrapositive, which is logically equivalent, is 'If a function is not continuous, then it is not differentiable.'

A student observes a function that is continuous over its entire domain but is not differentiable at x = 2. This observation serves as a counterexample to which of the following incorrect claims?

A) If a function is differentiable, then it is continuous.

B) If a function is continuous, then it may not be differentiable.

C) If a function is continuous, then it must be differentiable.

D) If a function is not continuous, then it is not differentiable.

Correct Answer: C

The observation shows a case where the 'if' part (continuity) is true, but the 'then' part (differentiability) is false. This directly disproves the claim that continuity is sufficient to guarantee differentiability.

For a function to be differentiable at a point, it is necessary for the function to first be...

A) discontinuous at that point.

B) defined, but not necessarily continuous at that point.

C) continuous at that point.

D) increasing at that point.

Correct Answer: C

Continuity is a necessary precondition for differentiability. A function cannot have a derivative at a point where it is not continuous. This is a direct application of the rule that differentiability implies continuity.

Let f be a function defined for all real numbers. Which of the following statements is guaranteed to be false?

A) f is continuous at x=5 and differentiable at x=5.

B) f is continuous at x=5 but not differentiable at x=5.

C) f is not continuous at x=5 but is differentiable at x=5.

D) f is not continuous at x=5 and not differentiable at x=5.

Correct Answer: C

The statement 'If a function is differentiable at a point, then it is continuous at that point' makes it impossible for a function to be differentiable at a point where it is not continuous. Therefore, statement C must be false.

Consider the properties of a function at a point x = c. Which statement provides the most accurate and complete description of the relationship between continuity and differentiability?

A) Differentiability is a necessary but not sufficient condition for continuity.

B) Continuity is a necessary but not sufficient condition for differentiability.

C) Differentiability is a sufficient but not necessary condition for continuity.

D) Continuity is both a necessary and sufficient condition for differentiability.

Correct Answer: B

Continuity is 'necessary' because a function must be continuous to be differentiable. It is 'not sufficient' because a function can be continuous without being differentiable (e.g., at a corner). Therefore, continuity is a necessary, but not sufficient, condition for differentiability.

If a function's graph has a break, jump, or hole at x = c, what can be concluded about the derivative at that point?

A) The derivative must exist and be equal to zero.

B) The derivative must exist but will be non-zero.

C) The derivative does not exist.

D) The derivative might exist, depending on the type of break.

Correct Answer: C

A break, jump, or hole signifies that the function is not continuous at x = c. Since a function must be continuous at a point to be differentiable there, the derivative cannot exist at x = c.