Math Problem Statement

Which of the following is statements is true?

If a function is continuous at a number A, then it is also differentiable at A

If a function is differentiable at a number A, then it is also continuous at A

If a function is differentiable at a number A, then it is also differentiable at any other number in its domain.

None of the statements above is true.

Solution

Let's analyze each statement to determine which one is true.

  1. If a function is continuous at a number AA, then it is also differentiable at AA.

    This statement is false. Continuity at a point does not imply differentiability. For example, the absolute value function f(x)=xf(x) = |x| is continuous at x=0x = 0, but it is not differentiable at x=0x = 0.

  2. If a function is differentiable at a number AA, then it is also continuous at AA.

    This statement is true. Differentiability at a point implies continuity at that point. If a function has a derivative at a point AA, it must also be continuous at AA.

  3. If a function is differentiable at a number AA, then it is also differentiable at any other number in its domain.

    This statement is false. Differentiability at one point does not imply differentiability at all points in the domain. For instance, the function f(x)=x1/3f(x) = x^{1/3} is differentiable at x=1x = 1, but it is not differentiable at x=0x = 0.

  4. None of the statements above is true.

    This statement is false because the second statement is true.

Conclusion: The true statement is:

"If a function is differentiable at a number AA, then it is also continuous at AA."

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

  1. Can you give an example of a function that is continuous but not differentiable?
  2. How can you prove that differentiability implies continuity?
  3. What are the conditions under which a function is differentiable at a point?
  4. Can you explain why differentiability at a point does not imply differentiability over the entire domain?
  5. What is the relationship between differentiability and the existence of a tangent line at a point?

Tip: Always remember that while differentiability implies continuity, the converse is not true. This is a fundamental concept in calculus.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Differentiability
Continuity
Calculus

Formulas

-

Theorems

Differentiability implies continuity

Suitable Grade Level

Grades 11-12, College Calculus