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AP Calculus AB 2.4 Connecting Differentiability and Continuity Question Bank

Practise AP Calculus AB 2.4 questions by testing continuity, corners, cusps and derivative limits at a point.

Syllabus
Effective Fall 2025
Course
AP Calculus AB

Exam points

  • use differentiability to infer continuity, but test derivative existence separately at corners or cusps
  • compare one-sided derivative limits and parameter values in piecewise functions

FUN-2.A—Explain the relationship between differentiability and continuity question 1

[Maximum number: 1]

If f(x)={x2 for x≤12x−1 for x>1f(x)=\left\{\begin{array}{ll}x^{2} & \text { for } x \leq 1 \\ 2 x-1 & \text { for } x>1\end{array}\right., then

A

f(x) is not continuous at x=1

B

f(x) is continuous at x=1 but f′(1)f^{\prime}(1) does not exist

C

f′(1)=2f^{\prime}(1)=2

D

lim⁡x→1f(x)\lim _{x \rightarrow 1} f(x) does not exist

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