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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- Mar 28th 2011, 12:57 PMDevenoTopological 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. - Mar 28th 2011, 01:37 PMOpalg
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 $\displaystyle x_n\to x$ then $\displaystyle f(x_n)\to f(x).$