Can I use the definition of continuity of function from Baby Rudin, setting X as empty set? Rudin does not specify X is a non-empty set but he supposes p is in X. Anyway if I use it for empty set X, then is a function with a domain E which is a subset of X continuous at p? One more extra question: If a definition contains two statements that contradict each other in its hypotheses, what happens? (I know if a theorem contains these things, then the theorem becomes vacuously true.) Note: for the Rudin's definition, please look at the attatched file.