The first instance of this is in the scene between Teiresias and Oedipus. Teiresias plainly says, "You mock my blindness? But I say you, with both your eyes, are blind." Oedipus, who saw plainly the riddle of the Sphinx, who is a great ruler over the city of Thebes, cannot see his own fate and his own life for what it is. Oedipus is, as Seth Benardete says, the totally public man, he can see the outside world but cannot see within himself, he cannot see the truth. This lacking of the 'inner' sight (private) is what gives Oedipus his gift with the 'outer' sight (public). He cannot encompass the both of them. When he does learn the truth, he blinds himself, thus destroying his 'public' persona, and since he is wholly public, he destroys himself. The Choragos tells Oedipus, "You were better off dead than alive and blind." .
The use of irony in Oedipus Rex reveals much about the character of Oedipus. We see that Oedipus truly is the 'public' man and can only possess one sight, that of the 'public' world. This public persona proved to be the end of him when he decreed his own fate to the people of Thebes. The 'private' sight, the inward sight, which Teiresias accuses Oedipus of lacking, is perhaps only suppressed in Oedipus, as he makes many ironic 'prophesies' of his own hinting that he knows of his true fate. .
The ultimate irony of the play is that Oedipus runs from Corinth for fear of the prophesy that he would murder his father and marry his mother coming true, and in running, he makes the prophesy come true. The Delphic Oracle told him that he would kill his father and marry his mother. To avoid this fate, Oedipus ran from whom he thought were his parents, and directly to whom were his real parents, where he did that which he set out to avoid. We see here the futility of trying to avoid the prophesies, another theme of the play. .
Jocaste says, "you will find no man who can give knowledge of the unknowable.