Which one of the following statements is/are correct?
$(a)$ Every bounded continuous real valued function on $\mathbb Q^2$ can be extended to a bounded continuous function on $\mathbb R^2.$
$(b)$ There exists a non-finite subset $X$ of $\mathbb Q^2$ such that every real valued continuous function on $X$ is bounded.
For $(a)$ I am thinking of a counter-example which is the following $:$
Consider the function $f : \mathbb Q^2 \longrightarrow \mathbb R$ defined by $$f(x,y) = \begin{cases} 2 & x \gt \sqrt 2 \\ 1 & x \lt \sqrt 2 \end{cases}$$ Then $f$ is continuous on $\mathbb Q^2$ but it has no continuous extension on $\mathbb R^2.$ So $(a)$ is definitely false. But I have no idea as to how to approach $(b).$ Any suggestion will be a boon for me at this stage.
Thanks for investing your valuable time in reading my question.