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.
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 $\displaystyle x_n\to x$ then $\displaystyle f(x_n)\to f(x).$