Continuity & inverse image of closed sets being closed
Question: Let be a function defined on a closed domain . Show that is continuous if and only if the inverse image of every closed set is a closed set.
One approach is to solve it using the fact that the complement of open sets are closed. However, I am wondering how I can prove it using the definition of a closed set using limit points?
I know that a set is closed if it contains all of its limit points. How could I use this definition to prove the above question? In other words, how do I show that if a set in the range holds all of its limit points, mapping it backwards to the domain will create a set also containing all of its limit points?
Thanks a lot!