Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a function. What conditions must $\varphi$ satisfy so that the metric space $(\mathbb{R},d_\varphi)$, where $d_\varphi(x,y)=|\varphi(x)-\varphi(y)|$, is complete? Of course, for $d_\varphi$ to be a metric, $\varphi$ must be an injective map.
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As you say, $\phi$ must be injective. Let its image be denoted $C$. By definition $\phi$ is an isometry between $\mathbb{R}$ with the metric $d_\phi$ and $C$ with the ordinary metric restricted from $\mathbb{R}$. Since completeness is invariant under isometry, it follows that $R$ is complete in the metric $d_\phi$ if and only if $C$ is complete as a subset of $\mathbb{R}$, which is true if and only if $C$ is a closed subset of $\mathbb{R}$.
Lee Mosher
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