According to the definition of continuity, given any $n_0\in \mathbb Z$, $\forall \epsilon\gt0$, $\exists\delta>0$ such that $\forall n \in \mathbb Z$ and $d_X(n,n_0)\lt \delta \Rightarrow d_Y\bigl(f(n),f(n_0)\bigl)\lt \epsilon$ holds when $\delta=\epsilon$. So I conclude that this graphically discrete function is continuous... Is there anything wrong?
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Since the space is discrete itself, this is not so weird. – Balloon Oct 15 '19 at 20:14
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Thank you guys! – Andy Z Oct 15 '19 at 20:14
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1Unless you put two different metrics on domain and codomain, it is continuous. However (I strongly believe) you still can come up with different $d_X, d_Y$ so that $1_{\mathbb{N}}$ is discontinuous. – Bumblebee Oct 15 '19 at 20:14
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See this – Bumblebee Oct 15 '19 at 20:17
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Every function $\mathbb Z\to X$ from $\mathbb Z$ with the Euclidean metric to another metric space $X$ is continuous. – Thomas Andrews Oct 15 '19 at 20:22
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You can always take $\delta=1$, as $d_X(n,n') < 1$ implies $n=n'$ for $n,n' \in \Bbb Z$ and then the $f$-values are the same.
Henno Brandsma
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