If $f:X \to \mathbb{R}$ is a continuous non constant function and $(X,d)$ is a connected metric space then $X$ is uncountable.
How to proceed this proof I am clueless , I know that if $X$ is connected so $f(X)$ is also connected and $f$ is non constant to $f(X)$ is an interval which is not singleton, also each interval is uncountable, so, $f(X)$ is uncountable does it means that $X$ is also uncountable? Any other methods other than this to understand this question?