Can there exist a continuous function from the real numbers to the rationals? Where we are considering the usually topology for the real numbers and the relative topology for the rationals with respect to the topology of real numbers. At first my intuition said no. Quickly going through random functions in my head, I couldn't create one. Is my intuition right? I've been thinking about a contradiction if a continuous function did exist, but can't find one that seals the deal. Can someone atleast confirm or correct my intuition without giving a solution? Hopefully I get a solution before tomorrow night. I'll have time at work to think about it some more, and try to use different approaches. I'll try looking at invariants and what not.