Is there an example of a non-compact and connected space $X$ such that constant functions are the only continuous functions from $X$ to $\mathbb{R}$ with the usual topology?

If the answer is "yes", I wonder under which conditions a non-compact and connected space possesses a non-constant continuous real-valued function.

Here are the related questions: Existence of non-constant continuous functions and Topological space $X$ which the set of non-constant real-valued continuous function on $X$ is empty.

Thank you.


Here is a easy example: An uncountable $X$ equipped with the cocountable topology is connected noncompact and the only continuous real-valued functions are constants. The proof is almost the same as the cofinite case.

Indeed, to admit a nonconstant $\mathbb{R}$-valued function means there is a nontrivial Hausdorff quotient $X\to f(X)$, and of course you can impose compactness there too since you can cut off $f$.

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    $\begingroup$ You can also see that if $f:X\to \Bbb R$ is continuous and $f(x) \ne f(y)$ then $f^{-1}(-\infty, (f(x)+f(y))/2)$ and $f^{-1}((f(x)+f(y)/2,\;\infty)$ are disjoint non-empty subsets of $X$, which is not possible ...........+1 $\endgroup$ – DanielWainfleet Sep 18 '18 at 19:24

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