Proposition 4

The oscillation $\omega(x; f)$ of an arbitrary function $f$, defined on a subset $X$ of a topological space $Y$ and taking its values in a metric space $X'$, is upper semi-continuous on $\overline{X}$.

Let $a$ be any point of $\overline{X}$; then for each $k > \omega(a; f)$ there exists an open neighbourhood $V$ of $a$ such that $\delta(f(V \cap X)) \leq k$; for each $x \in V \cap \overline{X}$, $V$ is a neighbourhood of $x$ and therefore

which shows that $\omega$ is upper semi-continuous at the point $a$.

In order that $\omega(x; f) = 0$ at a point $x \in \overline{X}$ it is necessary and sufficient that for each $\varepsilon > 0$ there should exist a neighbourhood $V$ of $x$ such that $f(V \cap X)$ is contained in a ball of radius $\varepsilon$; if $x \in X$, this condition expresses the fact that $f$ is continuous at the point $x$ (with respect to $X$); if $x \in \overline{X} \cap \mathbf{C}X$, the image under $f$ of the trace on $X$ of the neighbourhood filter of $x$ in $Y$ is a Cauchy filter base on $X'$; in particular: