Is the Derivative of a Differentiable Function Always Continuous?

  • Thread starter brum
  • Start date
In summary, the conversation discusses a problem involving a function and its derivative at x = 2. The first two statements must be true, but the third statement is false. Examples are given to support this conclusion, including a function with derivatives only at specific points and the Weierstrass function, which has a continuous derivative. Another example is mentioned, f(x) = (x^2) * sin(1/x), with f(0) = 0, which is differentiable at 0 but has a discontinuous derivative at 0.
  • #1
brum
81
0
Here is a problem that I am having trouble with.

Let f be a function such that:

Limit as h -->0 [f(2 + h) - f(2)] / h = 5

(essentially that: f ' (2) = 5 ).


It then asks: which of the following must be true?

1) f is continuous at x = 2.

2) f is differentiable at x = 2.

3) The derivative of f is continuous at x = 2.


THE ANSWERS: The first two are true, but the last one is false.

I understand the first two statements and how they must be true, but I also think the third statement must also be true. I can't think of an example of a function that disproves number 3. Can you?

I've tried drawing some possible functions for f ' that has a discontinuity (asymptote, jump, point) at x = 2 but whose antiderivative f is still continuous and differentiable at x = 2, but I can't come up with an example that works!

Thanks for any help
 
Physics news on Phys.org
  • #2
Unless I'm missing something, this doesn't seem like a simple problem. Consider a function that has derivatives only at 0, and points of the form 1/n, where n is a non-zero integer. Suppose its derivative at 0 is 0, and is 1 at all those other points. Then such a function has a derivative at 0, but it is discontinuous there, since any open set around 0 will contain some of the points of the form 1/n, but the derivatives there will always be 1, so the limit does not exist. Now, can a continuous function have such a weird "derivative"? Well, the Weierstrass Function is continuous but only has derivatives on a set of measure 0. The set consisting of 0 and 1/n for all non-zero integers n is a set of measure 0 (this is not a definition of measure 0, just an example of one). So there is a continuous function that has a derivative that is weird in some related sense, but that doesn't mean that it is "weird enough", and the Weierstrass function may still have a continuous derivative. Also, the discontinous derivative I suggested is one way it may be discontinuous, there may be other ways (like a jump discontinuity, etc.).
 
  • #3
I googled this up to the surface:

An example of a function which is differentiable at 0, but whose derivative is not continuous at 0

[itex]f(x) = (x^2) * sin(1/x), with f(0) = 0[/itex]

I haven't worked out the details. Perhaps someone else would like to try.
 
Last edited:

1. What is the purpose of "A question for you all"?

The purpose of "A question for you all" is to engage a group of people in a discussion and gather their opinions or perspectives on a specific topic.

2. Can anyone participate in "A question for you all"?

Yes, anyone can participate in "A question for you all". It is open to everyone, regardless of their background or expertise.

3. How often are new questions posted in "A question for you all"?

The frequency of new questions in "A question for you all" may vary depending on the platform or person posting them. However, it is common for new questions to be posted daily or weekly.

4. Are there any rules or guidelines for participating in "A question for you all"?

Some platforms or forums may have specific rules or guidelines for participating in "A question for you all". However, in general, it is important to respect others' opinions, be open-minded, and avoid any offensive or inappropriate language.

5. Can I suggest a question for "A question for you all"?

Yes, you can suggest a question for "A question for you all". This can be done by directly contacting the person or platform responsible for posting the questions or by participating in a discussion and suggesting a topic for future questions.

Similar threads

Replies
9
Views
1K
Replies
4
Views
223
Replies
20
Views
2K
  • Calculus
Replies
6
Views
1K
  • Calculus
Replies
1
Views
927
  • Calculus
Replies
14
Views
1K
Replies
3
Views
1K
Replies
4
Views
834
Replies
46
Views
3K
Back
Top