Topological space $X$ which every non-constant real-valued continuous function on $X$ is unbounded.

Question

Does there exists a topological space $X$ which every non-constant real-valued continuous function on $X$ is unbounded?

Answer 1

No, except in the vacuous case that there are no nonconstant continuous real-valued functions on $X$ at all. Indeed, suppose $f:X\to\mathbb{R}$ is nonconstant and continuous. Then composing $f$ with a homeomorphism $\mathbb{R}\to (0,1)$, we get a nonconstant continuous map $X\to(0,1)$, which is bounded when considered as a map $X\to\mathbb{R}$.

(Actually, you don't need anything so fancy as a homeomorphism $\mathbb{R}\to(0,1)$ for this argument. If you take two points $a<b$ in the image of $f$, you can instead just compose $f$ with the map $\mathbb{R}\to [a,b]$ which is the identity on $[a,b]$ and maps $(-\infty,a]$ to $a$ and $[b,\infty)$ to $b$.)