Prove that f is constant on [a,b]

Suppose that f: [a,b] -> R is continuous and that f([a,b]) is a subset of Q. Prove that f is constant on [a,b].

Suppose f is not constant. Then for some x1 and x2, f(x1) < f(x2).
Now between any two rational numbers there is an irrational q, ie. f(x1) < q < f(x2).
But since f is continuous, at some point p between x1 and x2, f(p) = q.
But f can't equal any irrational. So our assumption was wrong and f must be constant.