Let $(X,d)$ be a metric space, denote by $\tau_d$ the induced metric topology. Let $G_i$, $i\in I$, be a countable union of disjoint $G_\delta$-subsets. Mind that it could be that $G_i \notin \tau_d$ and $\cup_{i \in I}G_i$ could be a proper subset of $X$. Consider continuous maps $f_i:(G_i,d_{G_i}) \mapsto (Y,h)$, where $d_{G_i}$ is the restriction of $d$ to $G_i$ and $(G,h)$ is a metric space.
Question is it possible to patch together the maps $f_i$'s and extend the map so obtained, say $f$, to a continuous map $f:(X,d)\mapsto (Y,h)$?
Remark 1 Assume that in particular $G_i$'s are open and cover $X$, i.e. $X=\cup_{i \in I}G_i$. Then the disjoint union topological space $\coprod_{i\in I}(G_I,d_{G_i})$ and $(X,d)$ should coincide (since each ${\tau_{d_{G_i}}}$ would consists of intersections between $G_i$ and open sets, say $U$, of $\tau_d$; but then also $G_i\cap U \in \tau_d$). Then $f=f'$ would exists by the universal property of coproducts.
Remark 2 Let's now give up the assumption $X=\cup_{i \in I}G_i$, but retain $G_i \in \tau_d, \, \forall i \in I$. Endow $G=\cup_{i \in I}G_i$ with the restriction of $d$, say $d_G$. Then, we still have $G_i \in \tau_G$ and $f$ obtains as in the previous remark, but now as a continuous map from $(G,d_G)$ to $(Y,h)$. If $G$ is dense in $X$, then there exists also a continuous map $f'$ extending $f$ to $X$ (but it may not be unique, see e.g. the answer to this question Extending a function by continuity from a dense subset of a space ).
Remark 3 Let's finally give up also the assumption $G_i \in \tau_d, \, \forall i \in I$. If there existed $\tau_d$-open, disjoint sets $G_i'$ such that $G_i=G \cap G_i'$, then the $G_i \in \tau_{d_G}$, then it should be possible to obtain $f:(G,d_G)\mapsto (Y,h)$ as above and, if $G$ is again dense, then we can extend $f$ to some continuous $f'$.
Does the third remark make sense? Is it always possible to construct the aforementioned open sets $G_i'$? (If yes, this is not immediately clear to me). Can we obtain a positive answer to the above main question also beyond the restrictions in Remark 3? (That was the more general setting I managed to handle so far). Any help or refs are appreciated.
Best Answer
Every singleton set in $\mathbb R$ is a $G_{\delta}$ set. Consider the collection of sets $\{\{r\}: r \in \mathbb Q\}$. This is a disjoint countable collection of $G_{\delta}$ sets. Let $f$ be any map from $\mathbb Q$ to $\mathbb R$. This gives a family of continuous functions on these singletons. Your question asks if we can find a continuous map extending these functions. The answer is obviously NO.