Let $X$ be a topological space and $x\in X$. Then the quasi-component of the point $x$, denoted by $C_x$, is the intersection of all clopen (closed-and-open) subsets of $X$ which contain the point $x$. Clearly the quasi-components of two distinct points of a topological space $X$ either coincide or are disjoint, so the set of quasi-components constitutes a decomposition of the space $X$ into pairwise disjoint closed subsets. And clearly the connected component $C$ of a point $x$ in a topological space $X$ is contained in the quasi-component $C_x$ of the point $x$, and so every quasi-component is the union of the connected components of its points.

Question: Let $C_1$ and $C_2$ be two connected components of $X$ such that there exists $x\in X$ with $C_1\cup C_2\subseteq C_x$ and let $f$ be a real-valued, continuous function on $X$ such that $f(C_1)=\{0\}$. How can we show that $f(C_2)=\{0\}$?