Localization theorem

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 F(x) be a real-valued function defined on some domain Ω of the real line that is continuous in Ω. Let D be an arbitrary domain contained in Ω. The theorem states the following implication: $\int \limits _{D}F(x)dx=0~\forall D\in \Omega ~\Rightarrow ~F(x)=0~\forall x\in \Omega$

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