Bounded Function being absolutely integrable but not integrable

• jdz86
In summary, a bounded function is a function with a finite range and does not approach infinity as the input approaches infinity. A bounded function is absolutely integrable if the integral of its absolute value exists and is finite. This condition is stronger than regular integrability, where a function may be integrable but not absolutely integrable. It is possible for a bounded function to be absolutely integrable but not integrable, as seen in examples such as sin(1/x) and 1/x.
jdz86

Homework Statement

If f:[a,b] $$\rightarrow$$ $$\Re$$ is bounded then so is |f|, where |f|(x) = |f(x)|. Call f absolutely integrable if |f| is integrable on [a,b]. Give an example of a bounded function which is absolutely integrable but not integrable.

None

The Attempt at a Solution

I was thinking $$\sqrt{x}$$ but was unsure of how to show it.

How about looking for a nonintegrable function that takes only the values +1 and -1?

1. What is a bounded function?

A bounded function is a function that has a finite range or is limited between two values. In other words, the function's values do not approach infinity as the input approaches infinity.

2. What does it mean for a bounded function to be absolutely integrable?

A bounded function is absolutely integrable if the integral of the absolute value of the function exists and is finite. This means that the function is integrable over its entire domain and the area under the curve is finite.

3. How is absolute integrability different from regular integrability?

Absolute integrability is a stronger condition than regular integrability. A function may be integrable but not absolutely integrable if the integral of the function exists but the integral of the absolute value of the function does not exist or is infinite.

4. Can a bounded function be absolutely integrable but not integrable?

Yes, it is possible for a bounded function to be absolutely integrable but not integrable. This occurs when the integral of the function exists but the integral of the absolute value of the function does not exist or is infinite.

5. What are some real-world examples of bounded functions that are absolutely integrable but not integrable?

One example is the function f(x) = sin(1/x) on the interval [0, 1]. This function is bounded but not integrable since the integral of its absolute value does not exist. Another example is the function f(x) = 1/x on the interval [1, ∞). This function is absolutely integrable but not integrable since the integral of its absolute value is infinite.

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