Understanding Continuous Functions: Examining f'(7) Undefined

In summary, the conversation discusses whether a function f is continuous at x = 7 if its derivative is undefined. The options are a) f must be continuous at x = 7, b) f is definitely not continuous at x = 7, or c) there is not enough information to determine whether or not f is continuous at x = 7. The evidence points to c) as the correct answer, as there is no function mentioned that has no derivative at x = 7 and is not continuous. The conversation ends with Dan suggesting the function f(x) = \dfrac{1}{x - 7} as an example, to which the response is that this function would also be continuous at x = 7.
  • #1
bearn
11
0
Suppose f is a function such that f'(7) is undefined. Which of the following statements is always true? (Give evidences that supports your answer, then explain how those evidences supports your answer)

a. f must be continuous at x = 7.
b. f is definitely not continuous at x = 7.
c. There is not enough information to determine whether or not f is continuous at x = 7.
 
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  • #2
Can you think of a function f(x) where the derivative does not exist at x = 7 but is continuous there?

-Dan
 
  • #3
topsquark said:
Can you think of a function f(x) where the derivative does not exist at x = 7 but is continuous there?

-Dan
No,
 
Last edited:
  • #5
Oh, so the answer is a. f must be continuous at x=7?​
 
  • #6
bearn said:
Oh, so the answer is a. f must be continuous at x=7?​
Can you think of a function f(x) that has no derivative at x = 7 and is not continuous there?

-Dan
 
  • #7
I don't think there is
 
  • #8
bearn said:
I don't think there is
What about \(\displaystyle f(x) = \dfrac{1}{x - 7}\)?

-Dan
 
  • #9
topsquark said:
What about \(\displaystyle f(x) = \dfrac{1}{x - 7}\)?

-Dan
The answer should be C. then
 
  • #10
bearn said:
The answer should be C. then
Yes.

-Dan
 

1. What does it mean for f'(7) to be undefined?

When f'(7) is undefined, it means that the function f(x) is not differentiable at x = 7. This can happen for a variety of reasons, such as a sharp corner or a discontinuity in the graph of the function at x = 7.

2. Can a continuous function have an undefined derivative?

Yes, it is possible for a continuous function to have an undefined derivative. This occurs when the function has a sharp corner or a discontinuity at a certain point, which makes it non-differentiable at that point.

3. How can we determine if f'(7) is undefined?

To determine if f'(7) is undefined, we need to examine the graph of the function at x = 7. If the function has a sharp corner or a discontinuity at x = 7, then f'(7) will be undefined. We can also use the limit definition of the derivative to calculate f'(7) and see if it approaches a finite value or not.

4. What does it mean for a function to be continuous?

A function is continuous if it has no sudden jumps or breaks in its graph. This means that the graph of the function can be drawn without lifting the pen from the paper. Mathematically, a function f(x) is continuous at a point x = a if the limit of f(x) as x approaches a is equal to f(a).

5. Is it possible for a function to be continuous but not differentiable?

Yes, a function can be continuous but not differentiable. This occurs when the function has a sharp corner or a discontinuity at a certain point, making it non-differentiable at that point. However, it is also possible for a function to be continuous and differentiable at all points within a certain interval.

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