Corollary 1

Let $(X_i)_{i \in I}, (Y_i)_{i \in I}$ be two families of topological spaces with the same set of indices. For each $i \in I$, let $f_i$ be a mapping of $X_i$ into $Y_i$. In order that the product mapping $f : (x_i) \to (f_i(x_i))$ of

should be continuous at a point $a = (a_i)$ it is necessary and sufficient that $f_i$ is continuous at $a_i$ for each $i \in I$.

f can be written as $x \to (f_i(\mathrm{pr}_i(x)))$, so that by Proposition 1 the condition is sufficient. Conversely, for each $x \in I$ let $g_x$ be the mapping of $X_x$ into $\prod_{i \in I} X_i$ such that $\mathrm{pr}_x(g_x(x_x)) = x_x$ and $\mathrm{pr}_i(g_x(x_x)) = a_i$ whenever $i \neq x$; then $g_x$ is continuous at the point $a_x$, by Proposition 1. Since $f_x = \mathrm{pr}_x \circ f \circ g_x$ it follows that if $f$ is continuous at $a$, then $f_x$ is continuous at $a_x$.