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Math Help - Continuous Functions on Metric Spaces

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    Continuous Functions on Metric Spaces

    What is the definition of a continuous function on a metric space?

    And what are the basic properties?

    Thank you!
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  2. #2
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    Quote Originally Posted by lindsays View Post
    What is the definition of a continuous function on a metric space?

    And what are the basic properties?

    Thank you!

    Definition 1: a function f :X --> Y between metric spaces X,Y with metrix d_X , d_Y is continuous at x in X if for any e > 0 there exists h > 0 s.t. that d_X(x,w) < h ==> d_Y(f(x), f(w)) < e.

    Def. 2: a function f: X --> Y between metric spaces X,Y is continuous (in general) if the inverse image of any open ball in X is open in Y

    Def. 3: a function F: X --> Y betwen metrix spaces...and etc. is continuous at x in X if for any open ball B in Y containing f(x) there exists an open ball A in X containing x s.t. f(A) is contained in B.

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    Tonio
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    Quote Originally Posted by tonio View Post
    Definition 1: a function f :X --> Y between metric spaces X,Y with metrix d_X , d_Y is continuous at x in X if for any e > 0 there exists h > 0 s.t. that d_X(x,w) < h ==> d_Y(f(x), f(w)) < e.

    Def. 2: a function f: X --> Y between metric spaces X,Y is continuous (in general) if the inverse image of any open ball in X is open in Y

    Def. 3: a function F: X --> Y betwen metrix spaces...and etc. is continuous at x in X if for any open ball B in Y containing f(x) there exists an open ball A in X containing x s.t. f(A) is contained in B.
    To be clear: def 1 above is the one found in most textbooks introducing metric spaces.
    Def 2&3 are then proven a theorems.
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