Dictionary Definition
overspill
Noun
1 too much population [syn: overpopulation]
2 the occurrence of surplus liquid (as water)
exceeding the limit or capacity [syn: overflow, runoff] [also: overspilt]
Extensive Definition
In nonstandard
analysis, a branch of mathematics, overspill is a
widely used proof technique. It is based on the fact that the set
of standard natural numbers N is not an internal
subset of the internal set *N.
Indeed, by applying the induction principle for
the standard integers N and the transfer
principle we get the principle of internal induction:
For any internal subset A of *N, if

 1 is an element of A and
 for every element n of A, n+1 also belongs to A
 A= *N
If N were an internal set, then instantiating the
internal induction principle with N, it would follow N=*N which we
know not to be the case.
The overspill principle has a number of extremely
useful consequences:
 The set of standard hyperreals is not internal.
 The set of bounded hyperreals is not internal.
 The set of infinitesimal hyperreals is not internal.
In particular:
 If an internal set contains all infinitesimal nonnegative hyperreals, it contains a positive noninfinitesimal (or appreciable) hyperreal.
 If an internal set contains N it contains an unbounded element of *N.
Example
We can use these facts to prove equivalence of
the following two conditions for an internal hyperrealvalued
function f defined on *R.
 \forall \epsilon >\!\!\!> 0, \exists \delta >\!\!\!> 0, h \leq \delta \implies f(x+h)  f(x) \leq \epsilon
 \forall h \cong 0, \ f(x+h)  f(x) \cong 0
The proof that the second fact implies the first
uses overspill, since given a noninfinitesimal positive
ε
 \forall \mbox \delta \cong 0, \ (h \leq \delta \implies f(x+h)  f(x)
By overspill a positive appreciable δ
with the requisite properties exists.
These equivalent conditions express the property
known in nonstandard analysis as Scontinuity of f at x.
Scontinuity is referred to as an external property, since its
extension (e.g. the
set of pairs (f, x) such that f is Scontinuous at x) is not an
internal set.