What is an Integrable Function?
Generally speaking, if a function is integrable, all it means is that the integral is well defined and continuous.
Some types of functions with discontinuities , turns or other odd behavior may also be integrable— depending on the type and number of discontinuity. If the discontinuity is removable, then that function is still integrable. For example, the absolute value function y = |x| is integrable, even though x = 0 is undefined.
What is a Non Integrable Function?
Non integrable functions do not have well defined integrals. These may involve discontinuities that aren’t removable, like Oscillating Discontinuities or Essential Discontinuities (Second Type or Irremovable Discontinuity). Non integrable functions also include any function that jumps around too much, as well as any function that results in an integral with an infinite area.
Two simple functions that are non integrable are y = 1/x for the interval [0, b] and y = 1/x2 for any interval containing 0.
When mathematicians talk about integrable functions, they usually mean in the sense of Riemann Integrals. A Riemann integral is the “usual” type of integration you come across in elementary calculus classes. It’s the idea of creating infinitely small numbers of rectangles under a curve. However, there is another type of integration that can integrate even the most pathological of functions: Lebesgue Integration. This is usually tackled in more advanced mathematics classes.
Australian Mathematical Sciences Institute. Functions integrable and not. Retrieved November 23, 2019 from: https://amsi.org.au/ESA_Senior_Years/SeniorTopic3/3f/3f_5appendix_3.html
Bietcher, (2010). Integration – A Functional Approach. Springer Science & Business Media.
Stephanie Glen. "Integrable Function & Non Integrable Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/integrable-function-non-integrable-function/
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