Can there be a continuous non-constant function with only rational values defined on an interval in R? Or is the whole property meaningless?

If $f:[a,b]\to\mathbb{R}$ is a continuous function and $f(x)\in\mathbb{Q}$ for all $x\in[a,b]$ then what can say about $f$? My try: I think f should be constant, if it is not constant then it contradicts the continuity. Can anyone prove that f is constant?

Let $f\colon[0,1]\to\mathbb{R}$ be continuous such that $f(x)\in\mathbb{Q}$ for any $x\in[0,1]$. Intuitively I feel that $f$ is constant, since $\mathbb{Q}$ is dense in $\mathbb{R}$. How can I formally write this down?

The following is not a homework problem. I am doing it for self study. Prove that any continuous function from $\mathbb{R}$ to $\mathbb{Q}$ is constant. Here is my proof: Let $f:\mathbb{R}\rightarrow \mathbb{Q}$ be such a function. We first show that $\mathbb{Q}$ is disconnected. Let $p$...