Given the continuous maps $[0,\infty) \to \mathbb R$ define the following "truncation at level $K$ operator", $T$:

$T(f)(t) = f(\min(t, S_f))$, where $S_f = \inf \{ s : f(s) \ge K \}$

So essentially "$f$ is stopped when it hits level $K$". This kind of thing is very common in random process theory.

It follows from elementary classical analysis techniques that if $S$ is not a local maximum for $f$, and $g \to f$ uniformly then $T(g) \to T(f)$. The proof is straightforward but involves considering three or four cases. As such I was wondering if anyone has a reference for this that I can quote, or knows that this follows from some more general result. It should be something fairly standard from random process theory, and I guess the result even generalises to Skorohod space.