Question

Let f:[a,b]->Q be a continuous function. Prove that f is a constant function.

Answer 1

i would start by using a proof by contradiction
i.e. suppose it was not
then use the intermediate value theorem to show that since f is continuous, if \(f(x_1)=a, f(x_2)=b\) where a, b are rational and \(a\neq b\) then f must take on all values between \(a\) and \(b\) including the irrational ones

Answer 2

why use intermediate value theorem

Answer 3

because if f is continuous, and takes on two different values, then it must take on EVERY

Answer 4

value between the two. since between any two rationals there is an (infinite number) of irrationals, f must take on those as well, contradicting the assumption that f maps only to Q