Maurice Tutor

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    Argosy University/ Phoniex University/
    Nov-2005 - Oct-2011

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    Phoniex University
    Oct-2001 - Nov-2016

Category > Management Posted 04 Feb 2018 My Price 9.00

metric spaces

Is the function described in Example 5.4 uniformly continuous on X?

 

Example 5.4

 

Let (X1, d1) and (X2, d2) be metric spaces, let f: X1 → X2, and let c ∈ X1. Then the following three conditions are equivalent:

 

(a) f is continuous at c.

 

(b) If (xn) is any sequence in X1 such that (xn) converges to c, then (f(xn)) converges to f(c) in X2.

 

(c) For every neighborhood V of f(c) in X2, there exists a neighborhood U of c in X1 such that f(U) ⊆ V.

 

(a) ⇒ (b) Suppose f is continuous at c and let (xn) be a sequence in X1 such that xn → c. Given ε > 0, since f is continuous at c there exists

 

 

(c) ⇒ (a) Given any ε > 0, let V = N(f(c); ε). By our hypothesis in (c), there exists a neighborhood U = N(c; δ ) such that f (U ) ⊆ V. But then whenever d1(x, c) < δ="" we="" have="" x="">∈ U, so f (x) ∈ V and d2( f (x), f (c)) < ε.="" thus="" f="" is="" continuous="" at="">

 

 

 


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Maurice Tutor
(5)
Status NEW Posted 04 Feb 2018 07:02 PM My Price 9.00

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