Example of a non-integrable function f , such that |f| and f^2 are integrable?

In summary, the conversation discusses finding a function f:[a,b] -> R that is both integrable and discontinuous. Suggestions are given, including f(x) = 1 for x rational and -1 for x irrational. It is mentioned that this question may be better suited for the analysis forum.
  • #1
irresistible
15
0
I'm looking for a function
f:[a,b] -> R such that |f| and f2 are integrable on [a,b]
any helps?
 
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  • #2
To start with, f must be discontinuous (a continuous function on a closed interval is integrable). Can you think of a function f such that f is discontinuous, but |f| (and thus f2 = |f|2) is continuous?This is really an analysis question, so it really belongs in that forum. :)
 
Last edited:
  • #3
f(x)= 1 for x rational,
-1 for x irational
 
  • #4
Don't just give him an answer. ;)

(That's the exact same function I was thinking of, though.)
 
  • #5
I forgot to say: Only if you are dealing with Newton or rieman integrals, otherwise f(x) is integrable.
 
  • #6
Oh I don't know why I didn't think of that!
Thank you so much to both of youuuu!
:smile:
 

Related to Example of a non-integrable function f , such that |f| and f^2 are integrable?

1. What is an example of a non-integrable function f?

An example of a non-integrable function f is f(x) = 1/x on the interval [0, 1]. This function is not integrable because it has an infinite discontinuity at x = 0.

2. How can f(x) have an infinite discontinuity at x = 0?

Functions can have infinite discontinuities at certain points if they approach infinity or negative infinity at that point. In the case of f(x) = 1/x, as x approaches 0 from the positive side, the function approaches positive infinity. Similarly, as x approaches 0 from the negative side, the function approaches negative infinity.

3. Why are f(x) and f^2(x) both integrable in this example?

Despite f(x) not being integrable on its own, f^2(x) is integrable because the function becomes continuous at x = 0. This is because the negative and positive infinities cancel each other out when squared. Similarly, |f(x)| is integrable because it is essentially the absolute value of f(x) and the negative and positive infinities also cancel out.

4. Can a function be non-integrable at one point but integrable on a larger interval?

Yes, a function can be non-integrable at a specific point but still be integrable on a larger interval. This is because the integration process involves taking the limit of the function as the interval approaches the point in question. If the limit does not exist or is infinite, then the function is not integrable at that point. However, the function may still be integrable on a larger interval as the limit may exist and be finite within that interval.

5. Are there other examples of non-integrable functions where |f| and f^2 are integrable?

Yes, there are other examples of non-integrable functions where |f| and f^2 are integrable. One example is f(x) = sin(1/x) on the interval [0, 1]. This function has an infinite discontinuity at x = 0, making it non-integrable. However, |f(x)| = |sin(1/x)| is integrable on this interval, as is f^2(x) = sin^2(1/x).

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