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AP Calculus AB 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: May 2026

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

Question 1 of 9

According to the relationship between differentiability and continuity, if a function is known to be differentiable at a specific point, what other property must it also have at that same point?

All Questions (9)

According to the relationship between differentiability and continuity, if a function is known to be differentiable at a specific point, what other property must it also have at that same point?

A) It must be continuous.

B) It must be discontinuous.

C) It may or may not be continuous.

D) It must have a sharp corner.

Correct Answer: A

The provided content states that if a function is differentiable at a point, then it is continuous at that point. [cite: 1704]

A student observes that a function is continuous at x = c. Based solely on this information, what can be concluded about the function's differentiability at x = c?

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

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

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

D) The function must have a vertical tangent at x = c.

Correct Answer: C

The content specifies that a continuous function may fail to be differentiable at a point. This means continuity alone does not guarantee differentiability; it could be differentiable, or it could fail to be. [cite: 1707]

Which of the following statements is a correct summary of the relationship between differentiability and continuity?

A) Continuity implies differentiability, but differentiability does not imply continuity.

B) Differentiability implies continuity, but continuity does not imply differentiability.

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: B

This statement correctly combines the two main rules: 'If a function is differentiable at a point, then it is continuous' [cite: 1704] and 'A continuous function may fail to be differentiable' [cite: 1707], which means continuity does not guarantee differentiability.

Let f(x) be a function. If you are given that f(x) is differentiable at x = 5, which of the following MUST be true?

A) f(x) is continuous at x = 5.

B) f(x) is not continuous at x = 5.

C) f(x) may be differentiable at x = 6.

D) f(x) may not be continuous at x = 5.

Correct Answer: A

The core rule provided is that if a function is differentiable at a point, it is guaranteed to be continuous at that same point. [cite: 1704]

Consider the statement: 'A function can be continuous at a point within its domain but fail to be differentiable at that same point.' Based on the provided content, is this scenario possible?

A) No, because continuity is a stronger condition than differentiability.

B) No, because any continuous function is automatically differentiable.

C) Yes, this scenario is explicitly described as possible.

D) Yes, but only if the function is also discontinuous somewhere else.

Correct Answer: C

The provided content directly states, 'A continuous function may fail to be differentiable at a point in its domain.' This confirms that the described scenario is possible. [cite: 1707]

If a function g(x) is NOT continuous at a point x = a, what can be concluded about the differentiability of g(x) at x = a?

A) g(x) must be differentiable at x = a.

B) g(x) cannot be differentiable at x = a.

C) g(x) may or may not be differentiable at x = a.

D) More information is needed to determine differentiability.

Correct Answer: B

This question tests the contrapositive of the rule 'differentiability implies continuity.' If a function must be continuous to be differentiable, then if it is not continuous, it cannot possibly be differentiable. [cite: 1704]

Which statement best describes the hierarchy of function properties at a point?

A) Continuity is a necessary condition for differentiability.

B) Differentiability is a necessary condition for continuity.

C) The properties are equivalent; one implies the other.

D) The properties are unrelated.

Correct Answer: A

For a function to be differentiable, it must first be continuous. Therefore, continuity is a necessary prerequisite or condition for differentiability. [cite: 1704]

A function is known to have a derivative that does not exist at x = 2. What can be said about the function's continuity at x = 2?

A) The function must be continuous at x = 2.

B) The function must be discontinuous at x = 2.

C) The function could be continuous or discontinuous at x = 2.

D) The function must be equal to zero at x = 2.

Correct Answer: C

A function can fail to be differentiable for two main reasons related to this topic: it could be discontinuous, or it could be continuous but have a sharp corner or vertical tangent. Since both are possibilities, we cannot be certain about its continuity without more information. [cite: 1704, 1707]

The statement 'All continuous functions are differentiable' is false. Which piece of provided information serves as a direct counterexample to this claim?

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

B) A continuous function may fail to be differentiable at a point in its domain.

C) The relationship between differentiability and continuity can be explained.

D) A function that is not differentiable must also be not continuous.

Correct Answer: B

The claim is that continuity guarantees differentiability. The statement 'A continuous function may fail to be differentiable' directly contradicts this by stating that a function can be continuous without being differentiable, thus serving as the reason the original claim is false. [cite: 1707]