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The phrase "integrable function" is correct and usable in written English
It is typically used in mathematical contexts, particularly in calculus and analysis, to refer to a function that can be integrated over a specified interval. Example: "In order to apply the Fundamental Theorem of Calculus, we need to ensure that the function we are working with is an integrable function."
Dictionary
integrable function
noun
Any function that possesses a finite integral
Exact(60)
Definition C. Let be an integrable function, for One defines a linear functional as (2.22).
Definition B. Let be an integrable function; for one defines a linear functional as (2.1).
Let be a nonnegative integrable function with.
Let f be an integrable function on the unit disk.
where is a Lebesgue integrable function defined on because of.
Let f be an integrable function of period 2π.
We denote the set of all cone integrable function by.
for any integrable function f with compact support.
where is a given function and is an integrable function.
where g : [ 0, 1 ] → R is an integrable function.
Let (alpha>0), and (u(t)) be an integrable function.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com