Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions



(

f

n


)


{\displaystyle (f_{n})}
converges uniformly to a limiting function



f


{\displaystyle f}
on a set



E


{\displaystyle E}
if, given any arbitrarily small positive number



ϵ


{\displaystyle \epsilon }
, a number



N


{\displaystyle N}
can be found such that each of the functions




f

N


,

f

N
+
1


,

f

N
+
2


,
…


{\displaystyle f_{N},f_{N+1},f_{N+2},\ldots }
differ from



f


{\displaystyle f}
by no more than



ϵ


{\displaystyle \epsilon }
at every point



x


{\displaystyle x}
in



E


{\displaystyle E}
. Described in an informal way, if




f

n




{\displaystyle f_{n}}
converges to



f


{\displaystyle f}
uniformly, then the rate at which




f

n


(
x
)


{\displaystyle f_{n}(x)}
approaches



f
(
x
)


{\displaystyle f(x)}
is "uniform" throughout its domain in the following sense: in order to guarantee that




f

n


(
x
)


{\displaystyle f_{n}(x)}
falls within a certain distance



ϵ


{\displaystyle \epsilon }
of



f
(
x
)


{\displaystyle f(x)}
, we do not need to know the value of



x
∈
E


{\displaystyle x\in E}
in question — there can be found a single value of



N
=
N
(
ϵ
)


{\displaystyle N=N(\epsilon )}
independent of



x


{\displaystyle x}
, such that choosing



n
≥
N


{\displaystyle n\geq N}
will ensure that




f

n


(
x
)


{\displaystyle f_{n}(x)}
is within



ϵ


{\displaystyle \epsilon }
of



f
(
x
)


{\displaystyle f(x)}
for all



x
∈
E


{\displaystyle x\in E}
. In contrast, pointwise convergence of




f

n




{\displaystyle f_{n}}
to



f


{\displaystyle f}
merely guarantees that for any



x
∈
E


{\displaystyle x\in E}
given in advance, we can find



N
=
N
(
ϵ
,
x
)


{\displaystyle N=N(\epsilon ,x)}
(



N


{\displaystyle N}
can depend on the value of



x


{\displaystyle x}
) so that, for that particular



x


{\displaystyle x}
,




f

n


(
x
)


{\displaystyle f_{n}(x)}
falls within



ϵ


{\displaystyle \epsilon }
of



f
(
x
)


{\displaystyle f(x)}
whenever



n
≥
N


{\displaystyle n\geq N}
.
The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions




f

n




{\displaystyle f_{n}}
, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit



f


{\displaystyle f}
if the convergence is uniform, but not necessarily if the convergence is not uniform.

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