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Math Help - question on definition of continuity

  1. #1
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    question on definition of continuity

    if X is a function from metric space X to Y and x is an element of X, i need to show that the definition of continuity given by (for each open neighborhood V of f(x), there exists an open neighborhood U of x s.t. f(U) is a proper subset of V) coincides with the definition given by (whenever {X_n} is a sequence in X s.t. x_n converges to x, then f(x_n) converges to f(x)).

    i can't seem to relate the two definitions too well. help?
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by squarerootof2 View Post
    if X is a function from metric space X to Y and x is an element of X, i need to show that the definition of continuity given by (for each open neighborhood V of f(x), there exists an open neighborhood U of x s.t. f(U) is a proper subset of V) coincides with the definition given by (whenever {X_n} is a sequence in X s.t. x_n converges to x, then f(x_n) converges to f(x)).

    i can't seem to relate the two definitions too well. help?
    Let \varepsilon>0 and suppose for every open neighborhood V of f(x), there exists an open neighborhood U of x such that f(U) \subset V.

    Let \left\{x_n\right\} be a sequence in X such that x_n \rightarrow x. Then there is an N \in \mathbb{N} such that if n > N, x_n \in U implies f(x_n) \in f(U) \subset V meaning f(x_n) \rightarrow f(x).

    for the other direction, try to prove it by contradiction..
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