In Oedipus Rex, could Oedipus have avoided his fate?
The entire play is built around an inevitable fate for Oedipus. He is a character who was born to suffer this tragedy. In a sense, the play's intensity and tragic ending relies on the huge dramatic irony that exists: from the very beginning, the audience knows the true identity of Oedipus, but this is only something that he gradually becomes aware of before the final realisation. There is no sense therefore in which Oedipus could have avoided his fate. The seeds of his own destruction were planted long ago, and they are clearly going to burst into their flower during the course of the play. It is interesting that Jocasta seems to offer Oedipus a vision of life where fate does not control humans completely:
Fear? What should a man fear? It’s all chance, chance rules our lives. Not a man on earth can see a day ahead, groping through the dark. Better to live at random, best we can.
However, given Jocasta's own rising suspicions and her attempts to try and ignore the truth for as long as possible, it may be wiser to see this as Jocasta trying to convince herself as much as possible about a "truth" that she knows to be wrong. Oedipus is fated to realise who he is and what he has done, and this is a destiny he curses at the end of the play.