Is there an elementary way of proving that for any continuous function $f:\mathbb{Q}\to[0,1]$ there is such an $x\in\mathbb{R}\setminus\mathbb{Q}$ that $f$ can be extended to a continuous function $\mathbb{Q}\cup\{x\}\to[0,1]$, without resorting to the fact that $\mathbb{Q}$ is not a $G_\delta$-set in $\mathbb{R}$ and the
Theorem (4.3.20. in General Topology by Engelking): If $Y$ is a completely metrizable space, then every continuous mapping $f:A\to Y$ from a dense subset of a topological space $X$ to the space $Y$ is extendable to a continuous mapping $F:B\to Y$ defined on a $G_\delta$-set $B\subset X$ containing $A$.
which seem an overkill to me in this case?