# Partition a connected metric space

given two points p, q in a connected metric space and $\epsilon > 0$

I would like to show there exists pts $p=a_0,....,a_n=q$ s.t. the $d_X(a_i,a_{i-1})< \epsilon, \ \forall i\in{1,...,n}$

My idea is to say the distance function $d_X(p,x)$ is continuous and the continuous image of a connected set is connected. Then split up the range into $n$, $\epsilon$ sized intervals then take one pt from each interval and let a_i be the pre-image of that point.

I know $d_X(p,p)=0$ and $d_X(p,q) \geq 0$ so the distance must take on all the values in between.

So I think my only hangup is how to show the distance function is continuous.

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Triangle inequality is a good way to go. – mixedmath Dec 20 '11 at 1:39
Recall how topology (and thus continuity) is defined on a metric space (with balls). To show continuity of $f : x \mapsto d_X(x,p)$, take an open set $\Omega$ in $\mathbb{R}$ and check that $f^{-1}(\Omega)$ is open by taking an arbitrary point $x$ in it and finding an open ball centered at $x$ and contained in $f^{-1}(\Omega)$. – Joel Cohen Dec 20 '11 at 1:46

An alternative approach: Let $p$ and $\varepsilon>0$ be fixed. The set of $q$ such that there exists such a finite sequence of points is open and closed, and therefore is equal to the whole space.