$q\in Q×Q → f(q)=3$, so then $∀x∈R² → f(x)=3$

Question

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuous function on $\mathbb{R}$.
Also suppose that for every $q \in \mathbb{Q}\times\mathbb{Q}$ we have $f(q)=3$,
then show that: $\forall x \in \mathbb{R}^2$ we have $f(x)=3$.

Answer 1

Here are the steps to solve your problem:

1. Write down the definition for continuity of $f$ using quantifiers ($\forall,\,\exists$)
2. Substitute in the things you know into this, e.g how distances are measured in $\mathbb R^2$
3. Write down the one useful thing you know about the relationship between the reals and rationals (I.e. that for any real one can find an arbitrarily close rational)
4. Write it down with quantifiers
5. Write down what you want to prove
6. Stare really hard at it
7. Write down the proof