Weierstrass theorem states that for D from R^n that is compact and f: D --> R that is continuous function on D:
1) f is bounded from above and below;
2) f attains maximum and minimum on D;
3) for every minimizing and maximazing series Xk rho(Xk,D*) --> 0, where D* is the set of answers for minimizing/maximizing exercise.
My question is: which of these statements still hold when f is not continuos but semi-continuos from below?
Thanks for your help!
PS! Forgive me my English, I'm not a native speaker