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if $(X,d)$ is a connected metric space and there exist non-constant real-valued continuous function $f$ in $X$, show that $X$ is uncountable

pink floyd
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ann
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  • The hypothesis that there is a non-constant real-valued continuous function on $X$ ensures that $X$ has at least two points. – Brian M. Scott Apr 28 '15 at 18:35

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Hint: $f(X)$ is a connected subset of $\mathbb R$. But you don't have to assume existence of $f$: if $X$ has more than one point, try $f(x) = d(x,x_0)$ where $x_0 \in X$.

Robert Israel
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