What does integrability mean?
Integrability is the property that makes a function or mathematical object eligible for integration. In elementary calculus, it can mean that an antiderivative exists, so that the function can be represented by a derivative of another function. In real analysis, integrability more often concerns whether a definite integral can be assigned under a particular framework, such as Riemann or Lebesgue integration; continuous functions on closed intervals, for example, are Riemann integrable. Other mathematical contexts use the term for equations or systems that can be reduced to an integral form. The word is therefore closely associated with integration, but it names a condition rather than the process itself. Its importance lies in the many theorems that establish when apparently difficult sums of changing quantities have a precise mathematical limit, represented by an integral.
nounThe property of a function or mathematical object being integrable. In elementary calculus it may refer to the existence of an antiderivative; in analysis it more generally means that a suitable integral exists.
- The property of admitting an antiderivative, especially in elementary calculus.
- The property of having a well-defined integral under a specified mathematical framework, such as Riemann or Lebesgue integration.
- The property of a system or equation being reducible to, or expressible through, an integral formulation.
"The theorem guarantees the integrability of every continuous function on a closed interval."
"The theorem guarantees the integrability of every continuous function on a closed interval."
"Absolute integrability is required for several standard convergence theorems in Fourier analysis."
The word is normally treated as an uncountable abstract noun; the plural form is uncommon.
"The article contrasts the integrabilities of several different classes of functions."
Integrability asks whether a function's pieces can be brought together cleanly under the integral sign.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Where does integrability come from?
The word is formed from the English adjective integrable and the suffix -ity, which creates an abstract noun denoting a quality or state. Integrable ultimately comes through French from Latin integer, meaning whole or unbroken, a root also underlying integral, integer, integrity, and entire. In mathematical use, the terminology developed alongside the systematic calculus and analysis of functions.
Related word forms
How integrability is actually used
This is a specialized mathematical term, and its precise meaning depends on context: it may concern antiderivatives, the existence of definite integrals, or the integral formulation of an equation.
Easily confused with integrability
Integrability is the property that integration is possible, whereas integration is the process of combining a function's values under an integral.