Integrable function

Definition

An integrable function is a function for which the definite integral over a given interval exists and is finite. It must satisfy certain conditions, such as being bounded and having a limited number of discontinuities.

5 Must Know Facts For Your Next Test

1. A function is integrable on an interval $[a, b]$ if its absolute value has a finite integral over that interval.
2. The Riemann Integrability criterion states that a bounded function on a closed interval $[a, b]$ is integrable if the set of its discontinuities has measure zero.
3. Continuous functions on closed intervals are always integrable.
4. If a function is monotonic (either non-increasing or non-decreasing), it is also integrable.
5. Piecewise continuous functions with a finite number of discontinuities in $[a, b]$ are also integrable.

Review Questions

• What conditions must be met for a function to be considered integrable?
• How does the Riemann Integrability criterion determine if a function is integrable?
• Are all continuous functions on closed intervals necessarily integrable?

Related terms

Definite Integral: The definite integral of a function over an interval $[a, b]$ represents the net area between the function's graph and the x-axis within that interval.

$L^1$ Space: $L^1$ space consists of all functions whose absolute value's integral over their domain is finite. These functions are said to be absolutely integrable.

Riemann Sum: A Riemann sum approximates the area under a curve by summing up areas of rectangles whose heights are determined by the values of the function at specific points within subintervals.