Is the Indefinite Integral of a Riemann Integrable Function Always Continuous?

In summary, the continuity of an integral refers to the property of a function where small changes in the input result in small changes in the output, with no abrupt changes or breaks in its graph. To determine if an integral is continuous, the continuity property of integrals can be used, and checking for abrupt changes in the graph is also important. Continuity is important in integrals as it allows for evaluation and understanding of the integral function. A function can be continuous but not integrable, as seen in the example of f(x) = 1/x. The continuity of an integral is closely related to the continuity of the integrand function, with the continuity of the integral being dependent on the continuity of the integrand function.
  • #1
St41n
32
0
Suppose f: R -> R is integrable
Then, is F, the indefinite integral of f, a continuous function?
If this is not always true, what conditions do we need.
I know that if f is continuous, F is also continuous. What if f is a step function?
Can you think of any other interesting cases?

I'm searching all the internet and I can't find something that answers my question.
Thanks in advance
 
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  • #2
St41n said:
Suppose f: R -> R is integrable
Then, is F, the indefinite integral of f, a continuous function?
If this is not always true, what conditions do we need.
I know that if f is continuous, F is also continuous. What if f is a step function?
Can you think of any other interesting cases?

I'm searching all the internet and I can't find something that answers my question.
Thanks in advance

First I want to point out that there actually different types of integrals. For the integral that you are taught initially in a standard calculus sequence (the Riemann integral), the function must have certain properties that include continuity in a strict sense.

For a step function, you could the Riemann integral, but the function over your domain has to have these continuity properties, which means if you had say one discontinuity, you would have to split up your integral into two separate integrals each with its own limits and appropriate function.

If you want to integrate more general functions that are not Riemann integrable, then you have to use something like the Lebesgue integral. This is a more general version of integration that is based on a thing called measures (the study of measures is called measure theory). I should point out though that there are functions that are not Riemann integrable by Lebesgue integrable and also functions that are not Lebesgue integrable, but Riemann integrable.

A quick search for Riemann integrability gave this link:

http://www.math.cuhk.edu.hk/course/math2060a/Note 2. Riemann Integration.pdf
 
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  • #3
St41n said:
Suppose f: R -> R is integrable
Then, is F, the indefinite integral of f, a continuous function?
If this is not always true, what conditions do we need.
I know that if f is continuous, F is also continuous. What if f is a step function?
Can you think of any other interesting cases?

I'm searching all the internet and I can't find something that answers my question.
Thanks in advance

The integral of f is always continuous. If f is itself continuous then its integral is differentiable.

If f is a step function its integral is continuous but not differentiable.

A function is Riemann integrable if it is discontinuous only on a set of measure zero. So the function that is zero on the Cantor set and 1 on its compliment is Riemann integrable.
 
  • #4
So the integral of a Riemann integrable function is continuous. Thanks!
 

1. What is the definition of continuity of an integral?

The continuity of an integral refers to the property of a function where small changes in the input result in small changes in the output. In other words, the function is continuous if it has no abrupt changes or breaks in its graph.

2. How do you determine if an integral is continuous?

To determine if an integral is continuous, you can use the continuity property of integrals, which states that if a function is continuous on a closed interval, then it is also continuous on any subinterval of that interval. Additionally, you can check for any abrupt changes or breaks in the graph of the integral function.

3. What is the importance of continuity in integrals?

Continuity is important in integrals because it allows us to evaluate the integral using techniques such as the Fundamental Theorem of Calculus. It also helps us to understand the behavior of the integral function and make predictions about its values at different points.

4. Can a function be continuous but not integrable?

Yes, a function can be continuous but not integrable. For example, the function f(x) = 1/x is continuous on the interval [1, 2], but it is not integrable on this interval. This is because the function has a vertical asymptote at x = 0, which causes the integral to be undefined.

5. How does the continuity of an integral relate to the continuity of a function?

The continuity of an integral is closely related to the continuity of a function. In fact, the continuity of an integral is dependent on the continuity of the integrand function. If the integrand function is continuous, then the integral will also be continuous. However, if the integrand function is not continuous, then the integral may not be continuous.

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