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We improve Izumi's inequality, which states that any divisorial valuation centered at a closed point on a normal algebraic variety is controlled by the order of vanishing at . More precisely, as ranges through valuations that are monomial with respect to coordinates in a fixed birational model dominating , we show that for any regular function on at , the function is uniformly Lipschitz continuous as a function of the weight defining . As a consequence, the volume of is also a Lipschitz continuous function. Our proof uses toroidal techniques as well as positivity properties of the images of suitable nef divisors under birational morphisms.