Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
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
A function is integrable on an interval $[a, b]$ if its absolute value has a finite integral over that interval.
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.
Continuous functions on closed intervals are always integrable.
If a function is monotonic (either non-increasing or non-decreasing), it is also integrable.
Piecewise continuous functions with a finite number of discontinuities in $[a, b]$ are also integrable.
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.
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.