If $X$ is connected and if $f:X\rightarrow R$ is non-constant and continuous, then $X$ is uncountable.
Proof. Since $f$ is non-constant there are $a,b\in X $ such that without loss of generality $f(a)
- I feel like my proof isn't professional at all. Why is every non-degenerate interval uncountable? Would I just apply Cantor's Daigonalization?