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Definitions of continuity and convergence for functions defined on metric spaces.

Continuity for functions

Definition 1 A function f : ( X , d x ) ( Y , d y ) is continuous at a point x 0 X if: for any ϵ > 0 there exists a δ > 0 such that if d x ( x 0 , x 1 ) < δ , then d y ( f ( x 0 ) , f ( x 1 ) ) < ϵ .

Definition 2 A function: f : ( X , d x ) ( Y , d y ) is uniformly continuous if: for every ϵ > 0 there exists a δ > 0 such that for all x 0 X : if d x ( x 0 , x 1 ) < δ , then d y ( f ( x 0 ) , f ( x 1 ) ) < ϵ .

The difference between these definitions is that for a function to be simply continuous at a point, one need only find a δ for the given input x 0 , while for a function to be uniformly continuous, one needs to find for a given ϵ a single value of δ that works in the definition for every point over which the function is defined.

Example 1 Consider the function f : ( C [ T ] , d ) ( R , d 0 ) defined as f ( x ) = x ( t 0 ) . In words, the input to f is a continuous function over T , and the output from f is the value of the input function evaluated at t = t 0 . We ask the question: Is f continuous at some function input x 1 ( t ) ?

  • Designate a pair of functions x 1 ( t ) , x 2 ( t ) such that: d ( x 1 ( t ) , x 2 ( t ) ) < δ , where
    d ( x 1 ( t ) , x 2 ( t ) ) = sup t T | x 1 ( t ) - x 2 ( t ) | .
    In other words, s u p t T | x 1 ( t ) - x 2 ( t ) | < δ .
  • Next, we see that d 0 ( f ( x 1 ) , f ( x 2 ) ) = | x 1 ( t 0 ) - x 2 ( t 0 ) | .
  • Because | x 1 ( t 0 ) - x 2 ( t 0 ) | sup t T | x 1 ( t ) - x 2 ( t ) | , by the definition of the supremum, we can simply select δ = ϵ to get that if d 0 ( f ( x 1 ) , f ( x 2 ) ) < δ , then
    d 0 ( f ( x 1 ) , f ( x 2 ) ) d ( x 1 ( t ) , x 2 ( t ) ) < δ = ϵ .

This shows the continuity of f ( x ) at x 1 ( t ) . However, because the selection of δ did not depend on the value of x 1 , we have also shown that the function f is uniformly continuous.

Convergence of functions

Definition 3 The sequence of functions { f n } , f n : ( X , d x ) ( Y , d y ) converges pointwise to f : ( X , d x ) ( Y , d y ) if: for each x X , the sequence of values { f n ( x ) } converges to f ( x ) in ( Y , d y ) .

Definition 4 The sequence of functions { f n } , f n : ( X , d x ) ( Y , d y ) converges uniformly to f : ( X , d x ) ( Y , d y ) if: for each ϵ > 0 , there exists n 0 Z + such that if n n 0 , then d y ( f ( x ) , f n ( x ) ) < ϵ for all x X .

The difference between these definitions is that for uniform convergence, there must exist a single value of n 0 that works in the definition of continuity for all possible values of x X .

Example 2 Consider the sequence of functions x n ( t ) : ( [ 0 , 1 ] , d 0 ) ( [ 0 , 1 ] , d 0 ) given by x n ( t ) = t n . One naturally suspects that x n ( t ) may be converging to the zero-valued function. We can show this formally as follows:

  • Pick some t 0 , and check if { x 1 ( t 0 ) , x 2 ( t 0 ) , x 3 ( t 0 ) , x 4 ( t 0 ) , . . . } is converging to 0: Denote a n = x n ( t 0 ) = t 0 n , and notice that d 0 ( a n , 0 ) = | a n - 0 | = t 0 n . So, to pick an n 0 such that a n < ϵ if n n 0 , one only needs to note that since t 0 n t 0 n 0 , it then suffices for t 0 n 0 < ϵ , or in other words n 0 > t 0 ϵ . So this sequence converges pointwise to 0.
  • Additionally, because the range of possible inputs to x ( t ) is t [ 0 , 1 ] , we could select n 0 > 1 ϵ . Because 1 is the maximum value for t 0 , n 0 > 1 ϵ > t 0 ϵ will work for all values of t 0 [ 0 , 1 ] , and so the sequence { x n } also converges uniformly to the zero function.

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Graphene has a hexagonal structure
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Source:  OpenStax, Signal theory. OpenStax CNX. Oct 18, 2013 Download for free at http://legacy.cnx.org/content/col11542/1.3
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