Continuity and Differentiability

Topics in this chapter

• Limits and Continuity
• Differentiability and One-Sided Derivatives
• Derivative Rules and Techniques
• Special Function Derivatives
• Applications of Derivatives
• Mean Value Theorems

Cheat sheet

Practice questions

1. Question 1Q-ADE59F

   Let f(x) = (x^2 - 4)/(x - 2) for x ≠ 2 and f(2) = k. For which value of k is f continuous at x = 2?

   1. A.0
   2. B.2
   3. C.4
   4. D.Does not exist
   See the answer and solution
2. Question 2Q-107B7C

   The function f(x) = ax + b for x ≤ 1 and f(x) = x² for x > 1 is both continuous and differentiable at x = 1. What are the values of a and b?

   1. A.a = 1, b = 0
   2. B.a = 2, b = -1
   3. C.a = -2, b = 3
   4. D.a = 2, b = 1
   See the answer and solution
3. Question 3Q-E97F94

   Define f(x) = x² sin(1/x) for x ≠ 0 and f(0) = 0. Which statement is correct at x = 0?

   1. A.f is neither continuous nor differentiable
   2. B.f is continuous but not differentiable
   3. C.f is differentiable but not continuous
   4. D.f is both continuous and differentiable, but f′ is not continuous at 0
   See the answer and solution
4. Question 4Q-74C1D5

   Prove that if a function f is differentiable at x = a, then it is continuous at x = a.

   See the answer and solution
5. Question 5Q-9075FA

   Define f(x) = (1 − cos x)/x² for x ≠ 0 and f(0) = 1/2. What can be concluded about f at x = 0?

   1. A.f is both continuous and differentiable at x = 0
   2. B.f is continuous but not differentiable at x = 0
   3. C.f is differentiable but not continuous at x = 0
   4. D.f is neither continuous nor differentiable at x = 0
   See the answer and solution