Apply the definition of continuity with , and use the fact that there is a rational number in each open non empty interval.
Let f, g be continuous from to and suppose that f(r)=g(r) for all rational numbers r. Is it true that f(x)=g(x) for all .
I think yes, but am having trouble formulating a proof. Here is what I have so far.
Assume for contradiction that it is not true. Then , but from here I can't see where to get the contradiction. Any help?