Corollary 1

Let X, Y, E, B be topological spaces. Let $p: E\rightarrow B$ be an étale and separated mapping, let $h: X\times Y\rightarrow B$ be a continuous mapping and let $y_0$ be a point of Y. Let $g: X\times Y\rightarrow E$ be a mapping such that $p\circ g=h$. Suppose that the restrictions of $g$ to $X\times \{y_0\}$ on the one hand and, for every $x\in X$, to $\{x\} \times Y$ on the other hand, are continuous. If the space Y is connected and locally connected, the mapping $g$ is continuous.

Let us denote by Z the fibre product $(X\times Y)\times_BE$ and by $f: Z\rightarrow X\times Y,h': Z\rightarrow E$ the canonical projections. The mapping $f$ is étale (I, p. 31, prop. 8) and separated (I, p. 27, prop. 4). The mapping $s: X\times Y\rightarrow$ Z defined by $s(x, y) = ((x, y), g(x, y))$ is a section of $f$ satisfying the hypotheses of theorem 1. It is therefore continuous, as is $g=h'\circ s$.