= Influence function of the mean
{title2=$\operatorname{IF}(x;\mu,F)=x-\mu$}
For the <statistical functional> $T(F)=\mu=\mathbb E_F X$, assuming $\mathbb E_F|X|<\infty$, the contaminated <mean> is exactly
$$
T(F_{\varepsilon,x})=(1-\varepsilon)\mu+\varepsilon x.
$$
Thus the <influence function of the mean> is $x-\mu$, and its <gross-error sensitivity> is infinite on an unbounded sample space. The <linear approximation> is exact here: any fixed positive contamination fraction allows an arbitrarily large change by moving $x$ far enough. No finite second <moment> is needed for this <derivative>, although a finite <variance> is needed for the usual <central limit theorem> for the <sample mean>.
For comparison, the <influence function of the sample median> is
$$
\operatorname{IF}(x;\operatorname{Med},F)=\frac{\operatorname{sgn}(x-m)}{2f(m)},\qquad x\ne m,
$$
when the <median> $m$ is unique and the <probability density function> $f$ is positive and continuous at $m$. The <sign function> records which side of $m$ receives the contaminating mass. The <influence function of a quantile> supplies the short derivation. With the generalized-inverse <quantile> convention, contamination exactly at $x=m$ leaves the <median> at $m$, giving derivative zero there. If $f(m)$ is very small, this bounded <influence function> can still be large; if $f(m)=0$, the displayed formula is unavailable.
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