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

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 ∫_N F(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.

The theorem is easily generalized to multivariate functions, replacing intervals with the more general concept of connected open sets, that is, domains, and the original function with some F(x) : Rⁿ → R, with the constraints of continuity and nullity of its integral over any subdomain D ⊂ Ω. The proof is completely analogous to the single variable case, and concludes with the impossibility of finding a point x₀ ∈ Ω such that F(x₀) ≠ 0.

An example of the use of this theorem in physics is the law of conservation of mass for fluids, which states that the mass of any fluid volume must not change: $${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{f}}\rho ({\vec {x}},t)\,\mathrm {d} \Omega =0}$$

Applying the Reynolds transport theorem, one can change the reference to an arbitrary (non-fluid) control volume V_c. Further assuming that the density function is continuous (i.e. that our fluid is monophasic and thermodynamically metastable) and that V_c is not moving relative to the chosen system of reference, the equation becomes: $${\displaystyle \int _{V_{c}}\left[{{\partial \rho } \over {\partial t}}+\nabla \cdot (\rho {\vec {v}})\right]\,\mathrm {d} \Omega =0}$$

As the equation holds for any such control volume, the localization theorem applies, rendering the common partial differential equation for the conservation of mass in monophase fluids: $${\displaystyle {\partial \rho \over \partial t}+\nabla \cdot (\rho {\vec {v}})=0}$$