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.
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.
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.
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.
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.