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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|<var>F</var>(<var>x</var>)}} be a real-valued function defined on some open [[interval (mathematics)|interval]] <var>Ω</var> of the real line that is [[Continuous function|continuous]] in <var>Ω</var>. Let <var>D</var> be an arbitrary subinterval 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>
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