= Solution
A <statistical functional> is a map $T$ defined on a class of <probability distributions>. Its associated <functional statistic> is $T(F_n)$, where $F_n=n^{-1}\sum_i\delta_{X_i}$ is the <empirical distribution> and each $\delta_{X_i}$ is a <Dirac measure>. For example, $T(F)=\int x\,dF(x)$ yields the <sample mean>. The <influence function> records the <derivative> in the direction of point contamination:
$$
\operatorname{IF}(z;T,F)=\left.\frac{d}{d\varepsilon}T\big((1-\varepsilon)F+\varepsilon\delta_z\big)\right|_{\varepsilon=0+}.
$$
It is a local, infinitesimal robustness diagnostic, not a description of arbitrary large contamination.
Suppose $T$ is differentiable in a sense strong enough to linearize it at the <empirical distribution>, and that its <derivative> in a signed direction $G-F$ is $\int\operatorname{IF}(x;T,F)\,d(G-F)(x)$. Averaging the directions $\delta_z-F$ over $z\sim F$ gives the zero direction, so linearity implies $\int\operatorname{IF}\,dF=0$. The resulting <asymptotic linear representation> is
$$
T(F_n)-T(F)=\frac1n\sum_i\operatorname{IF}(X_i;T,F)+o_p(n^{-1/2}).
$$
If $\int\operatorname{IF}^2\,dF<\infty$, the <central limit theorem> and the negligible remainder give
$$
\boxed{\sqrt n\{T(F_n)-T(F)\}\ \xrightarrow{d}\ N\left(0,\int\operatorname{IF}(x;T,F)^2\,dF(x)\right).}
$$
Thus the <asymptotic variance> is $n^{-1}\int\operatorname{IF}^2\,dF$. For a vector functional the integral is $n^{-1}\int\operatorname{IF}\operatorname{IF}^{\mathsf T}\,dF$. Convergence of the actual finite-sample <variance> to this leading expression additionally requires control of second moments of the remainder, or an appropriate <uniform integrability> condition; <convergence in distribution> alone does not provide that. Existence of a contamination <derivative> by itself also does not guarantee the required empirical linearization.
Several complementary robustness measures follow. <Gross-error sensitivity> is $\gamma^*=\sup_z|\operatorname{IF}(z)|$. Under contamination $(1-\varepsilon)F+\varepsilon H$, the first-order bias is $\varepsilon\int\operatorname{IF}\,dH$, whose worst absolute coefficient is $\gamma^*$; bounded influence therefore bounds infinitesimal gross-error bias. The <local-shift sensitivity> is $\lambda^*=\sup_{x\ne y}|\operatorname{IF}(x)-\operatorname{IF}(y)|/|x-y|$, equal to $\sup|\operatorname{IF}'|$ when the <influence function> is smoothly differentiable. It measures the effect of moving a contaminant a small distance. The <rejection point of an influence function> is $\rho^*=\inf\{r:\operatorname{IF}(x)=0\text{ for }|x|>r\}$, using centered, standardized coordinates; it is infinite if there is no such cutoff. A finite cutoff completely rejects sufficiently extreme observations to first order. These measures address different aspects of robustness. The integral of squared influence controls asymptotic precision and efficiency, while <breakdown point> concerns finite amounts of contamination and is not determined by bounded influence alone.
For the <quantile>, assume $0<p<1$, a unique solution $q=q_p(F)$, and a <density> continuous and positive at $q$. Define <quantiles> of the contaminated distribution by generalized inversion, since a distribution with a point mass need not attain $p$ as an equality. If $z\ne q$, its indicator is constant near $q$ for sufficiently small contamination, and differentiating
$$
(1-\varepsilon)F(q_\varepsilon)+\varepsilon\mathbf1_{z\le q_\varepsilon}=p
$$
at zero gives $-p+f(q)q'_0+\mathbf1_{z\le q}=0$. Hence
$$
\boxed{\operatorname{IF}(z;q_p,F)=\frac{p-\mathbf1_{z\le q}}{f(q)}\quad(F\text{-almost every }z).}
$$
At the exceptional point $z=q$, the generalized-inverse <quantile> remains exactly $q$: below $q$ the contaminated <cumulative distribution function> is below $p$, and its value at $q$ exceeds $p$. Its actual <derivative> there is zero. The usual step-function expression is an almost-everywhere version, which is all that the <asymptotic variance> needs. Its mean is zero, and its second moment is
$$
\frac{p(p-1)^2+(1-p)p^2}{f(q)^2}=\frac{p(1-p)}{f(q)^2}.
$$
Therefore the <influence function of a quantile> gives <asymptotic variance> $p(1-p)/(nf(q)^2)$ and <gross-error sensitivity> $\max(p,1-p)/f(q)$. Its jump makes <local-shift sensitivity> infinite, and its nonzero constant tails make its rejection point infinite. If $f(q)=0$ or the <quantile> is not unique, this regular derivation does not apply.
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