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$8.\,\,\,$Let $d$ and $d'$ denote the usual and discrete metrics respectively on $\Bbb R$. Show that all functions $f$ from $\Bbb R$ with metric $d'$ to $\Bbb R$ with metric $d$ are continuous. What are the continuous function from $\Bbb R$ with metric $d$ to $\Bbb R$ with metric $d'$?

Now for the first part, I'm struggling to link the discrete and usual metric to show f is continuous and also I'm not sure what these continuous functions are? Maybe this comes clear after proving the first part....Thanks

Raul
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For the first part: the discrete metric induces the discrete topology, where every set is open. Hence, every function $f: (\mathbb R, d') \to (\mathbb R, d)$ is continuous, for the counter-image of every open set (in the topology induced by $d$) is open.

Now, the second part. Let $f: (\mathbb R, d) \to (\mathbb R, d')$ be a continuous function. In particular, for all $x \in \mathbb R$, $f^{-1}(x)$ is both closed and open in $(\mathbb R, d)$ (the singleton $\{ x\}$ is a closed and open set in the discrete topology). Now, $(\mathbb R,d)$, viewed as a topological space with the euclidean topology, is connected. Hence, for all $x \in X$, $f^{-1}(x)$ is either empty or the whole $\mathbb R$. It follows that $f$ is a constant function (because $f^{-1}(x)$ can't be empty for all $x \in \mathbb R$).

Francesco Genovese
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