= Sublinear path space for fluid queues
{title2=$(\mathcal C_0,\|\cdot\|_{\rm sl})$}
The useful infinite-horizon path space for a stable <queue> is
$$
\mathcal C_0=\{f\in C([0,\infty)):f(0)=0,\ f(t)/(1+t)\to0\},\qquad
\|f\|_{\rm sl}=\sup_{t\geq0}|f(t)|/(1+t).
$$
For $\delta,\sigma>0$, the workload functional $R(f)=\sup_{t\geq0}(\sigma f(t)-\delta t)$ is finite and <continuous>. Fix $f$, and choose $T\geq1$ so that $|\sigma f(t)|\leq\delta t/4$ for $t\geq T$. If $\|g-f\|_{\rm sl}<\delta/(4\sigma)$, then $\sigma g(t)-\delta t<0$ for $t\geq T$. Both suprema can thus be taken over $[0,T]$, where
$$
|R(g)-R(f)|\leq\sigma(1+T)\|g-f\|_{\rm sl}.
$$
Uniform convergence on bounded time intervals alone does not give this continuity: a pulse escaping to later times can produce a large workload while tending locally to zero.
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