=== Necessary Condition === Suppose $D$ is a connected domain. If $z, w \in D$, there exists a path $\gamma: \left[{0 \,.\,.\, 1}\right] \to D$ with $\gamma \left({0}\right) = z$ and $\gamma \left({0}\right) = w$. From the Paving Lemma, it follows that there exist $\epsilon \in \R_{>0}$ and a subdivision $\left\{ {x_0, x_1, \ldots, x_n }\right\}$ of $\left[{0 \,.\,.\, 1}\right]$ such that: :$\displaystyle \bigcup_{k \mathop = 0}^n B_\epsilon \left({\gamma \left({x_k}\right) }\right) \subseteq D$ and for all $k \in \left\{ {0, 1, \ldots, n - 1}\right\}$: :$\gamma \left({\left[{x_k \,.\,.\, x_{k + 1} }\right] }\right) \subseteq B_\epsilon \left({\gamma \left({x_k}\right) }\right)$ where $B_\epsilon \left({\gamma \left({x_k}\right) }\right)$ is the open $\epsilon$-ball about $\gamma \left({x_k}\right)$. Let $\operatorname{Re} \left({x_{k + 1} - x_k}\right)$ denote the real part of $x_{k + 1} - x_k$, and $\operatorname{Im} \left({x_{k + 1} - x_k}\right) $ denote the imaginary part of $x_{k + 1} - x_k$. For all $k \in \left\{ {0, 1, \ldots, n - 1}\right\}$, define the smooth paths $\gamma_{2k+1}, \gamma_{2k+2}: \gamma: \left[{0 \,.\,.\, 1}\right] \to D$ by: :$\gamma_{2k+1} \left({t}\right) = x_k + t \left({\operatorname{Re} \left({x_{k + 1} - x_k}\right) }\right)$ :$\gamma_{2k+2} \left({t}\right) = x_k + \operatorname{Re} \left({x_{k + 1} - x_k}\right) + i t \left({\operatorname{Im} \left({x_{k + 1} - x_k}\right) }\right)$ These are the line segments connecting $x_k$, $x_k + \operatorname{Re} \left({x_{k + 1} - x_k}\right)$, and $x_{k + 1}$. It follows that $x_k + \operatorname{Re} \left({x_{k + 1} - x_k}\right) \in B_\epsilon \left({\gamma \left({x_k}\right) }\right)$, as: {{begin-eqn}} {{eqn | l = \left\vert{\operatorname{Re} \left({x_{k + 1} - x_k}\right) }\right\vert       | o = \le        | r = \left\vert{x_{k + 1} - x_k}\right\vert       | c = Modulus Larger than Real Part }} {{eqn | o = <       | r = \epsilon }} {{end-eqn}} Then Open Ball is Convex Set shows that $\gamma_{2k+1} \left({t}\right), \gamma_{2k+2} \left({t}\right) \in B_\epsilon \left({\gamma \left({x_k}\right) }\right) \subseteq D$ for all $t \in \left[{0 \,.\,.\, 1}\right]$. It follows that the images of $\gamma_{2k+1}$ and $\gamma_{2k+2}$ are subsets of $D$.  Define $C_k$ as the directed smooth curve that is parameterized by $\gamma_k$. Thus $C_{2 k - 1}$ has start point $x_{k - 1}$, and $C_{2 k}$ has end point $x_k$. Define $C$ as the concatenation of $C_1, \ldots, C_{2n}$. Then $C$ is a staircase contour in $D$ with start point $z$ and end point $w$. :File:ConnectedDomainStaircase.png Illustration of the open balls inside the connected domain $D$. The path $\gamma$ between $w$ and $z$ is grey, and the constructed staircase contour $C$ is red. {{qed|lemma}} === Sufficient Condition === Suppose that for all $z, w \in \C$, there exists a staircase contour in $D$ with start point $z$ and end point $w$. Let $\gamma: \left[{a \,.\,.\, b}\right] \to D$ be a parameterization of $C$, where $\left[{a \,.\,.\, b}\right]$ is a closed interval. Then $\gamma$ is a path in $D$ with $\gamma \left({a}\right) = z$ and $\gamma \left({b}\right) = w$. Define $\gamma_0: \left[{0 \,.\,.\, 1}\right] \to D$ by $\gamma_0 \left({t}\right) = \gamma \left({a + t \left({b - a}\right) }\right)$. Then $\gamma_0$ is also a path in $D$ with $\gamma_0 \left({0}\right) = z$ and $\gamma_0 \left({1}\right) = w$. By definition of path-connected, it follows that $D$ is path-connected. Hence, $D$ is a connected domain. {{qed}}