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.




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

1. **If a function is continuous at a number \( A \), then it is also differentiable at \( A \).**

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

2. **If a function is differentiable at a number \( A \), then it is also continuous at \( A \).**

   This statement is **true**. Differentiability at a point implies continuity at that point. If a function has a derivative at a point \( A \), it must also be continuous at \( A \).

3. **If a function is differentiable at a number \( A \), 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 \( f(x) = x^{1/3} \) is differentiable at \( x = 1 \), but it is not differentiable at \( x = 0 \).

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 \( A \), then it is also continuous at \( A \)."**

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.

Which of the following is statements is true?

1. If a function is continuous at a number A, then it is also differentiable at A.
2. If a function is differentiable at a number A, then it is also continuous at A.
3. If a function is differentiable at a number A, then it is also differentiable at any other number in its domain.
4. None of the statements above is true.

Explore the relationship between differentiability and continuity in calculus. Find out which statement about continuous and differentiable functions is true. Suitable for advanced high school or college students studying calculus.

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