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Math Help - Topological counter-example (sequential continuity)

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    Topological counter-example (sequential continuity)

    hello math experts.

    i am looking for a non-continuous function f:X-->Y where X,Y are non-countable spaces, but where lim n-->∞ xn = x implies lim n-->∞ f(xn) = f(x).

    the simpler f, X,Y and the toplogies on X and Y are, the better.
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    Quote Originally Posted by Deveno View Post
    hello math experts.

    i am looking for a non-continuous function f:X-->Y where X,Y are non-countable spaces, but where lim n-->∞ xn = x implies lim n-->∞ f(xn) = f(x).

    the simpler f, X,Y and the toplogies on X and Y are, the better.
    Let X be an uncountable set with the cocountable topology, and let Y be the same set with the discrete topology. The identity function f from X to Y is discontinuous (the inverse image of an open set need not be open). But every convergent sequence in X is eventually constant, so if x_n\to x then f(x_n)\to f(x).
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