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In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the uniform limit theorem states that the
uniform limit of any sequence of
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s is continuous.
Statement
More precisely, let ''X'' be a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, let ''Y'' be a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, and let ƒ
''n'' : ''X'' → ''Y'' be a sequence of functions converging uniformly to a function ƒ : ''X'' → ''Y''. According to the uniform limit theorem, if each of the functions ƒ
''n'' is continuous, then the limit ƒ must be continuous as well.
This theorem does not hold if uniform convergence is replaced by
pointwise convergence
In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...
. For example, let ƒ
''n'' :
, 1nbsp;→ R be the sequence of functions ƒ
''n''(''x'') = ''x
n''. Then each function ƒ
''n'' is continuous, but the sequence converges pointwise to the discontinuous function ƒ that is zero on
, 1) but has ƒ(1) = 1. Another example is shown in the adjacent image.
In terms of function spaces, the uniform limit theorem says that the space ''C''(''X'', ''Y'') of all continuous functions from a topological space ''X'' to a metric space ''Y'' is a closed set">closed subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
of ''Y
X'' under the uniform metric. In the case where ''Y'' is Complete metric space, complete, it follows that ''C''(''X'', ''Y'') is itself a complete metric space. In particular, if ''Y'' is a Banach space, then ''C''(''X'', ''Y'') is itself a Banach space under the
uniform norm
In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when t ...
.
The uniform limit theorem also holds if continuity is replaced by
uniform continuity
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
. That is, if ''X'' and ''Y'' are metric spaces and ƒ
''n'' : ''X'' → ''Y'' is a sequence of uniformly continuous functions converging uniformly to a function ƒ, then ƒ must be uniformly continuous.
Proof
In order to prove the
continuity of ''f'', we have to show that for every ''ε'' > 0, there exists a
neighbourhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
''U'' of any point ''x'' of ''X'' such that:
:
Consider an arbitrary ''ε'' > 0. Since the sequence of functions ''(f
n)'' converges uniformly to ''f'' by hypothesis, there exists a natural number ''N'' such that:
:
Moreover, since ''f
N'' is continuous on ''X'' by hypothesis, for every ''x'' there exists a neighbourhood ''U'' such that:
:
In the final step, we apply the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of Degeneracy (mathematics)#T ...
in the following way:
:
Hence, we have shown that the first inequality in the proof holds, so by definition ''f'' is continuous everywhere on ''X''.
Uniform limit theorem in complex analysis
There are also variants of the uniform limit theorem that are used in complex analysis, albeit with modified assumptions.
Theorem.
Let
be an open and connected subset of the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. Suppose that
is a sequence of holomorphic functions
that converges uniformly to a function
on every compact subset of
. Then
is holomorphic in
, and moreover, the sequence of derivatives
converges uniformly to
on every compact subset of
.
Theorem.
Let
be an open and connected subset of the complex numbers. Suppose that
is a sequence of univalent
[Univalent means holomorphic and injective.] functions
that converges uniformly to a function
. Then
is holomorphic, and moreover,
is either univalent or constant in
.
Notes
References
* {{cite book
, author = James Munkres
, author-link = James Munkres
, year = 1999
, title = Topology
, edition = 2nd
, publisher =
Prentice Hall
Prentice Hall was a major American publishing#Textbook_publishing, educational publisher. It published print and digital content for the 6–12 and higher-education market. It was an independent company throughout the bulk of the twentieth cen ...
, isbn = 0-13-181629-2
* E. M. Stein, R. Shakarchi (2003). ''Complex Analysis (Princeton Lectures in Analysis, No. 2)'', Princeton University Press.
* E. C. Titchmarsh (1939). ''The Theory of Functions'', 2002 Reprint, Oxford Science Publications.
Theorems in real analysis
Topology of function spaces