[Math] Extending continuous function from a dense set

Question

If $X$ is a metric space and $Y$ a complete metric space. Let $A$ be a dense subset of $X$. If there is a uniformly continuous function $f$ from $A$ to $Y$, it can be uniquely extended to a uniformly continuous function $g$ from $X$ to $Y$. I was trying to think of an example of a pointwise continuous function from set a rational numbers $Q$ to real line $R$ which cannot be extended to a continuous function $R$ to $R$. But could not get anywhere.

Answer 1

$\displaystyle \sin\left({\frac{1}{x+a}}\right)$, $a$ irrational.