# 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.

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13. M

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36. ### Deceptive uniform convergence question

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41. ### A question about uniform convergence

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42. ### Uniform convergence integration

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45. ### Uniform Convergence of Power Series

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46. ### Uniform Convergence of Poisson Kernel on [-π, π] minus (-a, a)

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47. ### Uniform convergence and derivatives question

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48. ### Uniform Convergence of Fourier sine and cosine series

Homework Statement f(x)= {1, ‐1/2<x≤1/2} {0, ‐1<x≤ ‐1/2 or 1/2<x≤1} State whether or not the function's Fourier sine and cosine series(for the corresponding half interval) converges uniformly on the entire real line ‐∞<x<∞ Homework Equations The Attempt at a Solution...
49. ### Need to find if a sequence of functions has uniform convergence

Homework Statement f_{n} is is a sequence of functions in R, x\in [0,1] is f_{n} uniformly convergent? f = nx/1+n^{2}x^{2} Homework Equations uniform convergence \Leftrightarrow |f_{n}(x) - f(x)| < \epsilon \forall n>= n_{o} \inN The Attempt at a Solution lim f_{n} = lim...
50. ### Uniform convergence of a quotient

Homework Statement Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that f_n \to f and g_n \to g uniformly on [a,b]. Prove that \frac{1}{g_n} is defined for large n and \frac{f_n}{g_n} \to \frac{f}{g} uniformly on [a,b]. Show that this is...