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.
The function in question is not continuous at any particular point p, because there aren't any points. However, the functionis continuous on the empty set, because it had no points. Another way to think about it is if the function is discontinuous you have to be able to produce a point where continuity fails, and you can't I think you're confused by the definition of vacuously true. If two statements in a theorem contradict each other, the theorem can't be true