Uniform Continuity

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function
1
2
3
4
choose ϵ
determine δ
move
x
0
show labels
This Demonstration illustrates a theorem of analysis: a function that is continuous on the closed interval
[a,b]
is uniformly continuous on the interval.
A function is continuous if, for each point
x
0
and each positive number
ϵ
, there is a positive number
δ
such that whenever
|x-
x
0
|<δ
,
|f(x)-f(
x
0
)|<ϵ
. A function is uniformly continuous if, for each positive number
ϵ
, there is a positive number
δ
such that for all
x
0
, whenever
|x-
x
0
|<δ
,
|f(x)-f(
x
0
)|<ϵ
. In the first case
δ
depends on both
ϵ
and
x
0
; in the second,
δ
depends only on
ϵ
.
​

External Links

Continuous Function (Wolfram MathWorld)
Finite Limit at a Finite Point
Limit (Wolfram MathWorld)

Permanent Citation

Izidor Hafner
​
​"Uniform Continuity"​
​http://demonstrations.wolfram.com/UniformContinuity/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011