Proposition 6

Let $X_1, X_2, Y$ be three topological spaces, $f$ a mapping of the product space $X_1 \times X_2$ into $Y$. If $f$ is continuous at the point $(a_1, a_2)$ then the partial mapping $x_2 \to f(a_1, x_2)$ of $X_2$ into $Y$ is continuous at the point $a_2$.

For this mapping is the composition of $f$ and the mapping $x_2 \to (a_1, x_2)$; hence the result follows from Proposition 4.

Proposition 6 is often expressed by saying that a continuous function of two variables is continuous with respect to each of them separately.