 Solution

The product topology is generated by the basis of a topology of open rectangles $U\times V$, with $U$ open in $X$ and $V$ open in $Y$. For a fixed $y$,Both possibilities are open; checking this basis of a topology proves that $i_y$ is continuous. The analogous map $j_x:Y\to X\times Y$, $j_x(y)=(x,y)$, is continuous by the same argument.

If either factor is empty the product is empty and connected. Otherwise let $h:X\times Y\to \{0,1\}$ be continuous. For each $y$, $h\circ i_y$ is constant since $X$ is connected. Thus $h(x,y)=h(x_0,y)$ for any fixed $x_0\in X$. The right-hand side is $h\circ j_{x_0}$ and is constant in $y$ since $Y$ is connected. Therefore $h$ is constant on the whole product. The given characterization proves $X\times Y$ is connected.