  We show that the standard partition of unity subordinate to an open cover of
a metric space has Lipschitz constant $\max(1,M-1)/\mathcal{L}$, where
$\mathcal{L}$ is the Lebesgue number and $M$ is the multiplicity of the cover.
If the metric space satisfies the approximate midpoint property, such as length
spaces do, then the upper bound improves to $(M-1)/(2\mathcal{L})$. These
Lipschitz estimates are optimal. We also address the Lipschitz analysis of
$\ell^{p}$-generalizations of the standard partition of unity, their partial
sums, and their categorical products. Lastly, we characterize metric spaces
with Assouad--Nagata dimension $n$ as exactly those metric spaces for which
every Lebesgue cover admits an open refinement with multiplicity $n+1$ while
reducing the Lebesgue number by at most a constant factor.
