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''of a sequence of functions (mappings)''

A property of a sequence  $  f _ {n} :  X \rightarrow Y $, 
where  $  X $
is an arbitrary set,  $  Y $
is a metric space,  $  n = 1, 2 \dots $
converging to a function (mapping)  $  f:  X \rightarrow Y $, 
requiring that for every  $  \epsilon > 0 $
there is a number  $  n _  \epsilon  $(
independent of  $  x $) 
such that for all  $  n > n _  \epsilon  $
and all  $  x \in X $
the inequality

$$ 
\rho ( f ( x), f _ {n} ( x))  <  \epsilon
$$

holds. This is equivalent to

$$ 
\lim\limits _ {n \rightarrow \infty } \ 
\sup _ {x \in X } \ 
\rho ( f _ {n} ( x), f ( x))  =  0.
$$

In order that a sequence  $  \{ f _ {n} \} $
converges uniformly on a set  $  X $
to a function  $  f $
it is necessary and sufficient that there is a sequence of numbers  $  \{ \alpha _ {n} \} $
such that  $  \lim\limits _ {n \rightarrow \infty }   \alpha _ {n} = 0 $, 
as well as a number  $  n _ {0} $
such that for  $  n > n _ {0} $
and all  $  x \in X $
the inequality

$$ 
\rho ( f _ {n} ( x), f ( x))  \leq   \alpha _ {n}  $$

holds.

Example. The sequence  $  \{ f _ {n} ( x) \} = \{ x  ^ {n} \} $, 
$  n = 1, 2 \dots $
converges uniformly on any interval  $  [ 0, a] $, 
$  0 < a < 1 $, 
but does not converge uniformly on  $  [ 0, 1] $.

A necessary and sufficient condition for uniform convergence that does not use the limit function is given by the [[Cauchy criteria|Cauchy criterion]] for uniform convergence.

==Properties of uniformly-convergent sequences.==

1) If  $  Y $
is a normed linear space and two sequences of mappings  $  f _ {n} :  X \rightarrow Y $
and  $  g _ {n} :  X \rightarrow Y $
converge uniformly on  $  X $, 
then for any  $  \lambda , \mu \in \mathbf C $
the sequence  $  \{ \lambda f _ {n} + \mu g _ {n} \} $
also converges uniformly on  $  X $.

2) If  $  Y $
is a linear normed ring, if the sequence  $  f _ {n} :  X \rightarrow Y $, 
$  n = 1, 2 \dots $
is uniformly convergent on  $  X $
and if  $  g:  X \rightarrow Y $
is a bounded mapping, then the sequence  $  \{ gf _ {n} \} $
also converges uniformly on  $  X $.

3) If  $  X $
is a topological space,  $  Y $
is a metric space and if a sequence of mappings  $  f _ {n} :  X \rightarrow Y $, 
continuous at  $  x _ {0} \in X $, 
converges uniformly on  $  X $
to  $  f:  X \rightarrow Y $, 
then  $  f $
is also continuous at  $  x _ {0} $, 
that is,

$$ 
\lim\limits _ {x \rightarrow x _ {0} } \ 
\lim\limits _ {n \rightarrow \infty } \ 
f _ {n} ( x)  = \ 
\lim\limits _ {n \rightarrow \infty } \ 
f _ {n} ( x _ {0} )  = \ 
\lim\limits _ {n \rightarrow \infty } \ 
\lim\limits _ {x \rightarrow x _ {0} } \ 
f _ {n} ( x).
$$

The condition of uniform convergence of the sequence  $  \{ f _ {n} \} $
on  $  X $
is essential in this result, in the sense that there are sequences of numerical functions, continuous on an interval, that converge at all points to a function that is not continuous on the interval in question. An example is  $  f _ {n} ( x) = x  ^ {n} $, 
$  n = 1, 2 \dots $
on  $  [ 0, 1] $. 
Uniform convergence of a sequence of continuous functions is not a necessary condition for continuity of the limit function. However, if  $  X $
is a compact set,  $  Y $
is the set of real numbers  $  \mathbf R $
and if all functions in a sequence of continuous functions  $  f _ {n} :  X \rightarrow \mathbf R $
simultaneously increase or decrease at all points  $  x \in X $
and the sequence has a finite limit:

$$ 
\lim\limits _ {n \rightarrow \infty }  f _ {n} ( x)  =  f ( x),
$$

then in order that  $  f $
be continuous on  $  X $
it is necessary and sufficient that  $  \{ f _ {n} \} $
converges uniformly on that set. Necessary, and simultaneously sufficient, conditions for the continuity of the limit of a sequence of continuous functions in general are given in terms of [[Quasi-uniform convergence|quasi-uniform convergence]] of the sequence.

4) If a sequence of Riemann- (Lebesgue-) integrable functions  $  f _ {n} :  [ a, b] \rightarrow \mathbf R $, 
$  n = 1, 2 \dots $
converges uniformly on  $  [ a, b] $
to a function  $  f:  [ a, b] \rightarrow \mathbf R $, 
then this function is also Riemann- (respectively, Lebesgue-) integrable, for any  $  x \in [ a, b] $
one has

$$ \tag{* }
\lim\limits _ {n \rightarrow \infty } \ 
\int\limits _ { a } ^ { x }  f _ {n} ( t)  dt  = \ 
\int\limits _ { a } ^ { x }  f ( t)  dt  = \ 
\int\limits _ { a } ^ { x } 
\lim\limits _ {n \rightarrow \infty }  f _ {n} ( t)  dt,
$$

and the convergence of the sequence  $  \{ \int _ {a}  ^ {x} f _ {n} ( t)  dt \} $
to  $  \int _ {a}  ^ {x} f ( t)  dt $
is uniform on  $  [ a, b] $. 
Formula (*) has been generalized to the case of a [[Stieltjes integral|Stieltjes integral]]. If, however, a sequence of integrable functions  $  f _ {n} $, 
$  n = 1, 2 \dots $
on  $  [ a, b] $
only converges at each point of the interval to an integrable function  $  f $, 
then (*) need not hold.

5) If a sequence of continuously differentiable functions  $  f _ {n} :  [ a, b] \rightarrow \mathbf R $, 
$  n = 1, 2 \dots $
on  $  [ a, b] $
converges at some point  $  x _ {0} \in [ a, b] $
and if the sequence of derivatives  $  \{ df _ {n} /dx \} $
converges uniformly on  $  [ a, b] $, 
then the sequence  $  \{ f _ {n} \} $
also converges uniformly on  $  [ a, b] $, 
its limit is a continuously differentiable function on the interval and

$$ 
{
\frac{d}{dx }
 }
\lim\limits _ {n \rightarrow \infty }  f _ {n} ( x)  = \ 
\lim\limits _ {n \rightarrow \infty } \ 

\frac{df _ {n} ( x) }{dx }
 ,\ \ 
a \leq  x \leq  b.
$$

Let  $  X $
be a set and  $  Y $
a metric space. A family of functions (mappings)  $  f _  \alpha  :  X \rightarrow Y $, 
$  \alpha \in \mathfrak U $, 
with  $  \mathfrak U $
a topological space, is said to be uniformly convergent as  $  \alpha \rightarrow \alpha _ {0} \in \mathfrak U $
to the function (mapping)  $  f:  X \rightarrow Y $
if for every  $  \epsilon > 0 $
there is a neighbourhood  $  U ( \alpha _ {0} ) $
of  $  \alpha _ {0} $
such that for all  $  \alpha \in U( \alpha _ {0} ) $
and  $  x \in X $
the inequality

$$ 
\rho ( f( x), f _  \alpha  ( x))  <  \epsilon
$$

holds.

For uniformly-convergent families of functions there are properties similar to the above-mentioned properties of uniformly-convergent sequences of functions.

The concept of uniform convergence of mappings can be generalized to the case when  $  Y $
is a [[Uniform space|uniform space]], in particular, when  $  Y $
is a topological group.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Springer  (1975)</TD></TR></table>

====Comments====
The theorem that a monotone sequence of continuous functions converges uniformly to its [[pointwise limit]] if this limit is continuous, is known as [[Dini theorem|Dini's theorem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 75–78</TD></TR></table>