# Math Help - Topological counter-example (sequential continuity)

1. ## 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.

2. Originally Posted by Deveno
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 $f(x_n)\to f(x).$