Definitions:
Let $E$ be a subset of $X$. By an extension of a function $f: E \to \mathbb R$, I mean a function $\bar f: X \to \mathbb R$ such that $f = \bar f$ on $E$.
Question: For every continuous function $f: \mathbb Q \to \mathbb R$, does there exist an extension $\bar f: \mathbb R \to \mathbb R$ that is continuous at each $q \in \mathbb Q$?
In other words, can every continuous function on the rationals be extended to a function on the reals that is continuous at every rational point?