Remark 2

It is possible for all the partial mappings determined by a map $f : X_1 \times X_2 \to Y$ to be continuous without $f$ being continuous on $X_1 \times X_2$ (cf. Chapter IX, § 5, Exercise 23). \* For example if $f$ is the mapping of the real plane $\mathbf{R}^2$ into $\mathbf{R}$ defined by

and $f(0, 0) = 0$, then all the partial mappings are continuous; but $f$ is not continuous at $(0, 0)$, since $f(x, x) = 1/2$ if $x \neq 0$.

If $g$ is a mapping of $X_1$ into $Y$, continuous at a point $a_1$, then the mapping $(x_1, x_2) \to g(x_1)$ of $X_1 \times X_1 \to Y$ is continuous at all points $(a_1, x_2)$, for it is the composition of $g$ and the projection onto $X_1$.

The results of this subsection are easily extended to an arbitrary product $\prod_{i \in I} X_i$ of topological spaces by remarking that this product is homeomorphic to the product $\left( \prod_{i \in J} X_i \right) \times \left( \prod_{i \in K} X_i \right)$ for any partition $(J, K)$ of $I$ (no. 1, Proposition 2).