= Local-shift sensitivity
{title2=$\lambda^*(T,F)$}
The local-shift sensitivity of a scalar <statistical functional> is
$$
\lambda^*(T,F)=\sup_{x\ne y}\frac{|\operatorname{IF}(x;T,F)-\operatorname{IF}(y;T,F)|}{|x-y|}.
$$
It measures the sensitivity of the first-order contamination effect to moving the contaminating observation. If the <influence function> is continuously differentiable with bounded derivative, the <mean value theorem> gives $\lambda^*\le\sup_x|\operatorname{IF}'(x)|$, and taking $y\to x$ proves equality. A jump in the <influence function> makes $\lambda^*$ infinite, even when <gross-error sensitivity> is finite.
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