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:

∫
D
F
(
x
)
d
x
=
0
∀
D
⊂
Ω
⇒
F
(
x
)
=
0
∀
x
∈
Ω
A simple proof is as follows: if there were a point `x`_{0} within `Ω` for which `F`(`x`_{0}) ≠ 0, then the continuity of `F` would require the existence of a neighborhood of `x`_{0} in which the value of `F` was nonzero, and in particular of the same sign than in `x`_{0}. 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 `x`_{0} is part of the *open* set `Ω`, all neighborhoods of `x`_{0} smaller than the distance of `x`_{0} 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 `x`_{0} in `Ω` for which `F`(`x`_{0}) ≠ 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**) : **R**^{n}→**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 **x**_{0} ∈ `Ω` such that `F`(**x**_{0}) ≠ 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:

d
d
t
∫
V
f
ρ
(
x
→
,
t
)
d
Ω
=
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 thermodinamically metastable) and that `V`_{c} is not moving relative to the chosen system of reference, the equation becomes:

∫
V
c
[
∂
ρ
∂
t
+
∇
⋅
(
ρ
v
→
)
]
d
Ω
=
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:

∂
ρ
∂
t
+
∇
⋅
(
ρ
v
→
)
=
0