For an injective start, the work function is the Lipschitz extension of its injective restriction

Fix a metric space $M$, a number $k\ge1$ of servers, and an initial configuration $C_0$ that occupies $k$ distinct points. For a request sequence $\sigma$ write $\widehat w_\sigma=\widehat w(C_0;\sigma;\cdot)$ for the unordered work function, and for configurations $X,Y$ write $\lVert X-Y\rVert=\sum_i d(X_i,Y_i)$ for the labelled movement cost.

Statement. For every $\sigma$, every configuration $Y$ and every $\varepsilon>0$ there is an injective configuration $X$ with

Since the opposite inequality $\widehat w_\sigma(Y)\le\widehat w_\sigma(X)+\lVert X-Y\rVert$ is the Lipschitz property of the work function, the two together say

that is, the work function is the largest $1$-Lipschitz extension of its own restriction to injective configurations, and therefore carries no information at degenerate configurations beyond what it already carries at injective ones.

Role. The classical theory treats a configuration as a set of $k$ points, so degenerate configurations, which place two servers at a single point, do not occur in it. A formalization over maps $\mathrm{Fin}\,k\to M$ must deal with them, and the growth bounds of the work function are consequently available only in their injective form. This lemma is the bridge: any inequality between work functions that is proved at injective configurations and is preserved by the Lipschitz extension holds at every configuration.

The hypothesis on $C_0$ is necessary. On the uniform three-point space with $k=2$ and $C_0=(p,p)$, after the single request $p$ the work function vanishes at $(p,p)$, while every injective configuration lies at distance $1$ from it and has work-function value $1$; the right-hand side above equals $2$ and the statement fails by $2$. What survives for a degenerate start is the same statement with an additive slack equal to twice the distance from $C_0$ to the nearest injective configuration.

Formalization note. The error term $\varepsilon$ is present because on a general metric space the infimum over injective configurations need not be attained. The movement cost is the labelled one; this is the stronger reading, since the witness $X$ may be relabelled freely and workFnU is invariant under relabelling.