We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant max(1,M−1)/L, where L is the Lebesgue number and M is the multiplicity of the cover. If the metric space satisfies the approximate midpoint property, as length spaces do, then the upper bound improves to (M−1)/(2L). These Lipschitz estimates are optimal. We also address the Lipschitz analysis of ℓp-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.