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An influence-function representer is a centered square-integrable function representing derivatives of a statistical functional along all admissible score functions. Representers may differ by a function orthogonal to the statistical tangent space. This pathwise definition is distinct from defining an influence function solely by point-mass contamination paths.

For any estimator $T$ in a metric parameter space, $d(T,f{)}^2+d(T,g{)}^2\ge d(f,g{)}^2/2$ by the triangle inequality. Average the two risk functions and replace their probability density functions by their minimum. This proves $\max (R_f,R_g)\ge d(f,g{)}^2\int \min (p_f,p_g)/4$. The overlap equals one minus total variation distance. The same proof applies to squared error for a real-valued statistical functional of the parameter, without requiring that functional to be injective.