1. The problem statement, all variables and given/known data A kind person is helping me to self-study some aspects of topology, continuity, etc. He posed the following exercise for me, which I can't do, but he doesn't have time to write up the full solution. Ex: Show that a mapping [tex]f[/tex] between Hausdorff spaces is continuous if and only if the sequence [tex]f(x_0), f(x_1), \dots[/tex] converges to [tex]f(x)[/tex] for every sequence [itex]x_0, x_1, \dots[/itex] converging to [itex]x[/itex]. 2. Relevant equations 3. The attempt at a solution To show f is continuous, one must show that every open set in the range of f(x) must have an inverse image which is also an open set. I also know the definition of a Hausdorff space as a topological space in which any two points are contained in disjoint open sets. But that's about as far as I get. Does this theorem have a name that I could look up? Or can anyone give me some more hints about how to progress the proof? TIA.