Let $u(x,y)\to l$ as $(x,y)\to z_0=(x_0,y_0).$ How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$
Choose $\epsilon>0.$ Then $\exists~\delta>0$ such that $|u(x,y)-l|<\epsilon~\forall~(x,y)\in (B(z_0,\delta)-\{(x_0,y_0\})\cap D$ ($D$ being the domain of $u$).
Of course the domain of $u(x,y_0)$ is $D_1=\{x:(x,y_0)\in D\}$
Thus for all $x\in((x_0-\delta,x_0+\delta)-\{x_0\})\cap D_1, |u(x,y_0)-l|<\epsilon.$ This far is easy. But how can I say that $x_0$ is a limit point of $D_1?$