What is Uniform convergence: Definition and 162 Discussions
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.
View More On Wikipedia.org
(
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.
View More On Wikipedia.org
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Is such a theorem exists? (Uniform convergence).
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F
Uniform convergence of series of functions
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Uniform Convergence and integration
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Uniform Convergence Proof (new question)
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Pointwise Limit & Uniform Convergence of fn(x)=n^(2)e^(-nx)
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M
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Uniform Convergence of Series on (0,1)
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G
Definition of uniform convergence written as math symbols
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Proof involving sequences of functions and uniform convergence
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Question about pointwise convergence vs. uniform convergence
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Show Uniform Convergence of Sequence of Functions on Set X
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Proving Uniform Convergence of {fn} to f When f is Continuous
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Uniform Convergence of fn(x) in [0,1]?
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Uniform Convergence of fn: Does fn(x)=nx^2/1+nx Converge?
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A theorem on uniform convergence
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Uniform convergence of Fourier Series satisfying Lipschitz condition
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Uniform convergence and improper integration
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A simple question: uniform convergence of sequences
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Pointwise v. uniform convergence
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Pointwise v. uniform convergence
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Pointwise => uniform convergence when?
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What is the Uniform Convergence Problem in Stochastic Functions?
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Uniform convergence of a series of functions
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Uniform Convergence on an Equicontinuous Set
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Real Analysis - Uniform Convergence
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I badly in yet another uniform convergence problem
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Attempt at a rigorous (dis)proof of uniform convergence
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Pointwise vs. Uniform Convergence.
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Proving Uniform Convergence of fn to f on [a,b]
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S
Questions on uniform convergence
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Question on uniform convergence
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D
Showing uniform convergence (or lack of) on [0,1]
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Uniform Convergence and Boundedness: Exploring the Relationship
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Pointwise vs Uniform Convergence
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T
Equivalent characterization of uniform convergence
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Prove Uniform Convergence of f^{(n)} to 0 on (-1,1)
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Uniform Convergence: Does f^-1 Have the Same Property?
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Uniform Convergence: Showing Continuous & Uniformly Continuous
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S
Does x^n f(x) Converge Uniformly on [0,1] as n Approaches Infinity?
Suppose f(x) is continuous on [0,1], and that f(1)=0. Prove that x^n f(x) converges uniformly on [0,1] as n \rightarrow \infty By continuity, if |x-1|< \delta then |f(x)|< \epsilon for x \in [x_0 ,1] for some x_0 \in [0,1]. And there is some N such that if n>N, then |x^n|<\epsilon since...