Localization theorem: Difference between revisions

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In [[mathematics]], particularly in [[integral calculus]], the '''localization theorem''' allows, under certain conditions, to infer the nullity of a function given only information about its continuity and the value of its integral.

Let {{math|F(<var>x</var>)}} be a real-valued function defined on some open [[Domain (mathematics)#Real and complex analysis|___domaininterval]] <var>Ω</var> of the real line that is [[Continuous function|continuous]] in <var>Ω</var>. Let <var>D</var> be an arbitrary ___domainsubinterval contained in <var>Ω</var>. The theorem states the following implication:

<math>\int\limits_D F(x) dx = 0 ~ \forall D \subset \Omega ~ \Rightarrow ~ F(x) = 0 ~ \forall x \in \Omega</math>

[[File:Localization Theorem.svg|float|right]]

A simple proof is as follows: if there were a point <var>x₀</var> within <var>Ω</var> for which {{math|F(<var>x₀</var>)&ne;0}}, then the continuity of {{math|F}} would require the existence of a [[neighborhood (mathematics)|neighborhood]] of <var>x₀</var> in which the value of {{math|F}} was nonzero, and in particular of the same sign than in <var>x₀</var>. Since such a neighborhood <var>N</var>, which can be taken to be arbitrarily small, must however be of a nonzero width on the real line, the integral of {{math|F}} over <var>N</var> would evaluate to a nonzero value. However, since <var>x₀</var> is part of the ''open'' set <var>Ω</var>, all neighborhoods of <var>x₀</var> smaller than the distance of <var>x₀</var> to the frontier of <var>Ω</var> are included within it, and so the integral of {{math|F}} over them must evaluate to zero. Having reached the contradiction that {{math|&int;_<var>N</var>F(<var>x</var>)dx}} must be both zero and nonzero, the initial hypothesis must be wrong, and thus there is no <var>x₀</var> in <var>Ω</var> for which {{math|F(<var>x₀</var>)&ne;0}}.

The theorem is easily generalized to [[multivariate calculus|multivariate function]]s, replacing intervals with the more general concept of connected open sets, that is, [[Domain (mathematics)#Real and complex analysis|___domain]]s, and the original function with some {{math|F(<var>'''x'''</var>) : '''R'''ⁿ&rarr;'''R'''}}, with the constraints of continuity and nullity of its integral over any subdomain {{math|<var>D</var>&sub;<var>Ω</var>}}. The proof is completely analogous to the single variable case, and concludes with the impossibility of finding a point {{math|<var>'''x'''₀</var> &isin; <var>Ω</var>}} such that {{math|F(<var>'''x'''₀</var>)&ne;0}}.

[[Category:Mathematics]]