= Solution
Population <influence functions> describe infinitesimal <contamination (statistics)>; finite-sample analogues examine actual changes to an empirical sample. For its <empirical distribution> $F_n$, add an observation $z$ and define the <sensitivity curve>
$$
\operatorname{SC}_n(z)=(n+1)\left[T\left(\frac n{n+1}F_n+\frac1{n+1}\delta_z\right)-T(F_n)\right].
$$
Under suitable differentiability it approximates the <influence function>. A replacement version is $\operatorname{RC}_{n,i}(z)=n[T_n(y_1,\ldots,z,\ldots,y_n)-T_n(y)]$, approximately $\operatorname{IF}(z;T,F_n)-\operatorname{IF}(y_i;T,F_n)$. Their supremum magnitudes quantify finite one-observation sensitivity.
The <empirical influence function> instead substitutes $F_n$ directly in $\operatorname{IF}(z;T,F)$. For a differentiable <M-estimator> this gives $-\psi(z,T(F_n))/[n^{-1}\sum_i\partial_\theta\psi(y_i,T(F_n))]$. For quantile functionals, a discrete unsmoothed empirical law may not admit the required density derivative; a <sensitivity curve> or a smoothed empirical law is then preferable.
Global contamination is measured by <finite-sample maximum bias> $b_n(m;y)=\sup_{z:d_H(z,y)\leq m}|T_n(z)-T_n(y)|$, where $d_H$ counts replacements. Under the convention of the first fraction that causes unbounded bias,
$$
\boxed{\varepsilon_n^*(T;y)=\frac1n\min\{m:b_n(m;y)=\infty\}.}
$$
This is the finite-sample <replacement breakdown point>. The largest replacement fraction still guaranteed bounded is one grid step smaller, so the conventions must be specified. One replacement sends a <sample mean> arbitrarily far, giving $1/n$, while the ordinary scalar <sample median> has first-breaking fraction $\lceil n/2\rceil/n$ with the usual midpoint convention for even samples. Its limiting value is $1/2$. Local influence and global breakdown answer different questions; a bounded <influence function> is not by itself a proof of a high breakdown point.
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