What instances of dramatic irony appear in Oedipus Rex?
Well, the whole play is structured around what, to its original audience, would be one colossal case of dramatic irony. Sophocles' audience would have already known the Oedipus story, and the very name "Oedipus" would be synonymous with sleeping with his mother and killing his father. That means that, even before the play began, the audience would know the ending. This terrific dramatic irony would mean that, every time Oedipus talks about finding the cause of the Theban plague, the audience would know exactyl what the cause was: Oedipus himself.
Within the construction of the play, there are ironies all over the place. As Oedipus killed his father on the crossroads, he felt confident that he'd left his father behind him in Corinth. Oedipus solves the riddle of the Sphinx (focussing on the way that time ages and weakens men) but the crown of Thebes which he wins causes him to become aged and weakened in just the same way.
Oedipus is determined to find out the truth, and seek the murderer so that he can see him and confront him. Of course, Oedipus does find the truth and the murderer - but he can't see him - because he is Oedipus. Oedipus' response is to blind himself.
Even Oedipus' name is an irony: it means "swollen-footed" or "I think I know", meaning that both Oedipus' origins as the Theban heir and his self-assured insistence on knowing are written tragically into his very name from the first moment of the play.
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The entire play is based on dramatic irony. Oedipus goes to another land in order to escape his fate from killing his father and marrying his mother but in fact goes to the same land he was born in. Not only that, along the way he himself commits these crimes unknowingly. Oedipus tries to find who murdered the late King but in the path of trying to find out the truth, is ultimately going to his own demise. He is the killer. He is actually bound by his fate more then he has free will.