This problem was given in one of these math contests my school had and that no one was able to prove it (in extreem detail). Problem: Suppose that f is a function with the property that lim f(x) exists for all real c x->c Define a new function g this way: g(x) = lim f(t) x->t Prove in hideous detail that g is continuous everywhere, (i.e. continuous at every real number). As far as i can tell you, i m not at this level of proving such a problem, but it would be helpful if someone could explain it to me on doing this problem. As it would help me understand my class related topics better. This is was a contest question, so I dont expect everyone being able to do it, as even my teacher said it is difficult.