Proving that if $U \cap C \neq \varnothing$ then $U \subseteq C.$ 
Let $(X, \tau)$ be a topological space and let $C \subseteq X$ be a connected component. Then for any connected set $U \subseteq X$ with $U \cap C \neq \varnothing$ show that $U \subseteq C.$

I am thinking along the following lines $:$
We write $U = (U \cap C) \cap (U \cap C^c).$ If $U \cap C^c \neq \varnothing$ I am trying to show that then $U$ would become disconnected. But for that we need to show that both $U \cap C$ and $U \cap C^c$ are clopen subsets of $U,$ which I can't quite able to conclude. Could anyone please give me some suggestion in this regard?
Thanks for investing your valuable time in reading my question.
EDIT $:$ What will happen if we draw two disjoint circles in the complex plane and join their centres by a straight line? Then the straight line will cut both the circles at a single point.

If you define the component $C(x)$ as the maximal connected subset of the space that contains $x$, then by $C\cap U \neq \emptyset$, there exists $x\in U\cap C$ and since $U$ is connected it implies that $U\subseteq C$.
