Corollary 1

Let X be a topological space and Y a uniform space. The set of mappings of X into Y which are continuous at a point $x_0 \in X$ is closed in $F_u(X; U)$.

If V is the neighbourhood filter of $x_0$ in X, $u(x_0)$ is a cluster point of $u(V)$; hence $u$ is continuous at $x_0$ if and only if $u(V)$ is a Cauchy filter base on Y (Chapter II, § 3, no. 2, Proposition 5, Corollary 2).