Is the anti-derivative of a continuous function continuous?

In summary, the conversation discusses the problem of proving the differentiability of the anti-derivative of a continuous function and the use of the fact that the integral of a continuous function is continuous in the proof. The definition of an anti-derivative is also mentioned, where it is stated that an anti-derivative is always at least as "smooth" as the original function. Ultimately, the conversation ends with a proposed proof for the assertion and gratitude for the help provided.
  • #1
Niles
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Hi guys

I have been wondering: Say we have a continuous function f. I integrate f to obtain its anti-derivative called capital f, i.e. F. Now I wish to prove the differentiability of F, and in order to do so, I need the fact that F is continuous (this is just something I need in my proof).

Now, the problem is that I cannot deduce continuity of F on the fact that F is differentiable, since I wish to prove the differentiability of F. What can I do instead?

I thought of using the argument (which I am not sure is correct) that the integral of a continuous function is continuous. Am I allowed to do this?
 
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  • #2
Niles said:
I thought of using the argument (which I am not sure is correct) that the integral of a continuous function is continuous. Am I allowed to do this?

How would you argue for it? :smile:

Try to make a proof for that assertion!


By the way, yes, it is provably correct..:smile:
 
  • #3
Hmm, I can't come up with a proof for it, but I mean: Why shouldn't it be continuous? If we look at it as describing the area under a continuous graph, then needless to say, it has to be continuous. But again, this is hardly a proof..
 
  • #4
Isn't the fact that f is integratable (to F) enough to show that F is differentiable?
 
  • #5
I am told to do it "the long way" :smile:
 
  • #6
There's something important you can say about the anti-derivative. What is it? The answer is in the name.
 
  • #7
Werg22 said:
There's something important you can say about the anti-derivative. What is it? The answer is in the name.

I cannot use that F is an anti-derivative (i.e. that it is differentiable) to prove its continuity, since I am trying to prove its differentiability. So I must use another argument.
 
  • #8
If F is the antiderivative of f then F'=f, so obviously F is in C^1.
 
  • #9
I am quoting what I said earlier:


Niles said:
I cannot use that F is an anti-derivative (i.e. that it is differentiable) to prove its continuity, since I am trying to prove its differentiability. So I must use another argument.
 
  • #10
The whole point of being an "anti-derivative" is that it has a derivative! How else are you going to use the fact that this function is the anti-derivative of some function?
 
  • #11
HallsofIvy said:
The whole point of being an "anti-derivative" is that it has a derivative! How else are you going to use the fact that this function is the anti-derivative of some function?

But isn't that a circular argument? I mean, I am trying to prove differentiability, so I can't argue from this "ansatz"/claim that it is continuous.
 
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  • #12
Alright, then, how are you defining "anti-derivative"? After all, if you are going to prove anything about an anti-derivative, you are going to have to use the fact that is is an anti-derivative- you are going to have to use the definition.

The definition I know of "anti-derivative" is "F(x) is an anti-derivative of f(x) if and only if F'(x)= f(x)". It pretty much follows from the fact that f(x) is the derivative of F(x) that F(x) is differentiable. And, of course, "differentiable" is stronger than "continuous". If F(x) is differentiable, it must be continuous.

The anti-derivative is a "smoothing" operation. F(x) is always at least as "smooth" as f(x). Even if f(x) is not continous, F(x) is. And F(x) is "continuously differentiable" wherever f(x) is continous.
 
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  • #13
HallsofIvy said:
Alright, then, how are you defining "anti-derivative"?

I would to it like you did. But you have a good point.

Thanks to all for helping.
 
  • #14
since the integral over a interval [x,x] is always zero (given the integrand is continuous)
then take the limit of the next expression:

[tex] F(x+h)-F(x)=\int^{x+h}_{0}f(x)dx-\int^{x}_{0}f(x)dx=\int^{x+h}_{x}f(x)dx[/tex]
 

1. What is an anti-derivative?

An anti-derivative is the reverse process of differentiation. It is a function that, when differentiated, gives the original function back.

2. What is a continuous function?

A continuous function is a function that has no breaks or gaps in its graph. This means that the function can be drawn without lifting the pen from the paper.

3. Why is the anti-derivative of a continuous function important?

The anti-derivative of a continuous function allows us to find the original function from its derivative, which is useful in many real-life applications, such as physics, economics, and engineering.

4. Is the anti-derivative of a continuous function always continuous?

Yes, the anti-derivative of a continuous function is always continuous. This is because the anti-derivative is the reverse process of differentiation, and differentiation does not introduce any breaks or gaps in the function.

5. How can we prove that the anti-derivative of a continuous function is continuous?

We can use the Fundamental Theorem of Calculus to prove that the anti-derivative of a continuous function is continuous. This theorem states that if a function is continuous on an interval, then the area under its curve can be found by evaluating its anti-derivative at the endpoints of the interval. This implies that the anti-derivative must also be continuous on the same interval.

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