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[[File:Localization Theorem.svg|float|right]]
A simple proof is as follows: if there were a point <var>x</var><sub>0</sub> within <var>Ω</var> for which {{math|<var>F</var>(<var>x</var><sub>0</sub>) ≠ 0}}, then the continuity of {{math|<var>F</var>}} would require the existence of a [[neighborhood (mathematics)|neighborhood]] of <var>x</var><sub>0</sub> in which the value of {{math|<var>F</var>}} was nonzero, and in particular of the same sign than in <var>x</var><sub>0</sub>. 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|<var>F</var>}} over <var>N</var> would evaluate to a nonzero value. However, since <var>x</var><sub>0</sub> is part of the ''open'' set <var>Ω</var>, all neighborhoods of <var>x</var><sub>0</sub> smaller than the distance of <var>x</var><sub>0</sub> to the frontier of <var>Ω</var> are included within it, and so the integral of {{math|<var>F</var>}} over them must evaluate to zero. Having reached the contradiction that {{math|
The theorem is easily generalized to [[multivariate calculus|multivariate function]]s, replacing intervals with the more general concept of connected [[open set]]s, that is, [[Domain (mathematics)#Real and complex analysis|___domain]]s, and the original function with some {{math|<var>F</var>(<var>'''x'''</var>) : '''R'''<sup>''n''</sup>→'''R'''}}, with the constraints of continuity and nullity of its integral over any subdomain {{math|<var>D</var>⊂<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'''</var><sub>0</sub> ∈ <var>Ω</var>}} such that {{math|<var>F</var>(<var>'''x'''</var><sub>0</sub>) ≠ 0}}.
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