Proposition 9

Tag 0BE6, permanent. Cite it as [Bourbaki, Tag 0BE6].

Let $f$ be a uniformly continuous mapping of a product uniform space $X_1 \times X_2$ into a uniform space $Y$ ; then every partial mapping

x_2 \to f(x_1, x_2)

of $X_2$ into $Y$ is uniformly continuous.

In other words, a uniformly continuous function of two arguments is uniformly continuous with respect to each of them separately.

\* The example given in Chapter I, § 4, no. 2, Remark 2, shows that the converse of this proposition is false. \*

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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