Let $X$ be a non-empty and topological space. The set of all real-valued continuous function and all non-constant real-valued continuous function on $X$ is denoted by $C(X)$ and $NC(X)$, respectively.
Clearly, $C(X)\not=\emptyset$ since every constant function on $X$ is continuous. Also, if $|X|=1$, then $NC(X)=\emptyset$.
Does there exists a topological space $X$ with $|X|>1$ and non-trivial, which $NC(X)=\emptyset$. In the other words, does there exists a topological space $X$ with $|X|>1$ such that every real-valued continuou functions on $X$ is constant?