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

[St41n]

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

 

[chiro]

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

 

[lavinia]

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.

 

[St41n]

So the integral of a Riemann integrable function is continuous. Thanks!