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We study the influence function of the transport quantile map $\\mathbf{Q}_P$, defined as the optimal transport map pushing a fixed reference measure $\\mu$ forward to a target distribution $P$. For the Huber contamination $P_t=(1-t)P+t\\delta_{x_0}$, we prove that the first-order limit $\\mathbf{I}(x_0;\\mathbf{Q}_P(z)) := \\lim_{t\\downarrow 0} [\\mathbf{Q}_{(1-t)P+t\\delta_{x_0}}(z)-\\mathbf{Q}_P(z)]/t$ exists whenever $x_0\\ne \\mathbf{Q}_P(z)$ and characterize it uniquely. 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We study the influence function of the transport quantile map $\\mathbf{Q}_P$, defined as the optimal transport map pushing a fixed reference measure $\\mu$ forward to a target distribution $P$. For the Huber contamination $P_t=(1-t)P+t\\delta_{x_0}$, we prove that the first-order limit $\\mathbf{I}(x_0;\\mathbf{Q}_P(z)) := \\lim_{t\\downarrow 0} [\\mathbf{Q}_{(1-t)P+t\\delta_{x_0}}(z)-\\mathbf{Q}_P(z)]/t$ exists whenever $x_0\\ne \\mathbf{Q}_P(z)$ and characterize it uniquely. 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