Art & Science of Math
Rationals, irrationals, and continuity

• More information on complex exponentiation.
• An example of a function continuous at all irrationals, but discontinuous at all rationals. This page is part of a collection of calculus pathologies.
• To prove that there is no function that is continuous on the rationals but discontinuous on the irrationals requires the Baire Category theorem. See, for example, this pdf file or this discussion.