Solution to exercice 2.2a.16 of "Robust Statistics: The Approach Based on Influence Functions" On page 180 of Robust Statistics: The Approach Based on Influence Functions one finds the following question:

  
*
  
*16: Show that for location-invariant estimators always     $\varepsilon^*\leq\frac{1}{2}$. Find the corresponding upper bound on 
  the finite-sample breakdown point $\varepsilon^*_n$, both in the case 
  where $n$ is odd or $n$ is even.
  

The second part (after the period) is actually trivial (given the first) 
but I can't find a way to prove the first part (sentence) of the question.
In the section of the book pertaining to this question one finds (p98):

Definition 2: The finite-sample breakdown point $\varepsilon^*_n$ of an
  estimator $T_n$ at the sample $(x_l,\ldots, x_n)$ is given by:
  $$\varepsilon^*_n(T_n;x_i,\ldots,x_n):=\frac{1}{n}\max\{m:\max_{i_1,\ldots,i_m}\sup_{y_1,\ldots,y_m}\;|T_n(z_1,\ldots,z_n)|<\infty\}$$
where the sample $(z_1,\ldots,z_n)$ is obtained by replacing $m$ data
  points  $x_{i_1},\ldots,x_{i_m}$ by arbitrary values
  $y_1,\ldots,y_m.$

The formal definition of $\varepsilon^*$ itself runs for almost a page, but can be thought of as 
$$\varepsilon^*=\underset{n\rightarrow\infty}{\lim}\varepsilon^*_n$$ 
Although not defined explicitly, one can guess that location-invariant means that $T_n$ must satisfy
$$T_n(x_1,\ldots,x_n)= T_n(x_1+c,\ldots,x_n+c), \text{ for all } c\in \Bbb{R}$$

I (try to) answer whuber's question in the comment below. The book defines estimator $T_n$ is several pages, starting at p82, I try  to reproduce the main parts (I think it will answer whuber's question):

Suppose we have one-dimensional observations $(X_1,\ldots,X_n)$ which
  are  independent and identically distributed (i.i.d.). The
  observations belong to some sample space $\mathcal{H}$, which is a
  subset of the real line $\mathbb{R}$ (often $\mathcal{H}$ simply
  equals $\mathbb{R}$ itself, so the observations may take on any
  value). A parametric model consists of a family of probability
  distributions $F_\theta$,  on the sample space, where the unknown
  parameter $\theta$ belongs to some parameter space $\Theta$

...

We identify the sample $(X_1,\ldots,X_n)$ with its empirical 
  distribution $G_n$, ignoring the sequence of the observations (as is 
  almost always done). Formally, $G_n$, is given by 
  $(1/n)\sum_{i=1}^n\Delta_{x_i}$ where $\Delta_{X}$, is the point mass 
  1 in $X$. As estimators of $\theta$, we consider real-valued 
  statistics $T_n=T_n(X_1,\ldots,X_n)=T_n(G_n)$.  In a broader sense, an
  estimator can be viewed as a sequence of statistics $\{T_n,n\geq 1\}$
  , one for each possible sample size $n$.  Ideally, the observations
  are i.i.d. according to a member of the  parametric model
  $\{F_\theta;\theta\in\Theta\}$ , but the class
  $\mathcal{F}(\mathcal{H})$ of all possible probability distributions
  on  $\mathcal{H}$  is much larger.
We consider estimators which are functionals [i.e., $T_n(G_n)=T(G_n)$
  for all $n$ and $G_n$] or can asymptotically be replaced by 
  functionals. This means that we assume that there exists a functional 
  $T:\mbox{domain}(T)\rightarrow\mathbb{R}$ [where the domain of $T$ is 
  the set of all distributions $\mathcal{F}(\mathcal{H})$ for which $T$ 
  is defined] such that
  $$T_n(X_1,\ldots,X_n)\underset{n\rightarrow\infty}{\rightarrow}T(G)$$
  in probability when the observations are i.i.d. according to the true
  distribution $G$ in $\mbox{domain}(T)$. We say that $T(G)$ is the
  asymptotic value of  $\{T_n;n\geq 1\}$ at $G$.

...

In this chapter, we always assume that the functionals under study are
  Fisher consistent (Kallianpur and Rao, 1955):
  $$T(F_\theta)=\theta\;\mbox{ for all } \theta\in\Theta$$
  which means that at the model the estimator $\{T_n;n\geq 1\}$
  asymptotically measures the right quantity. The notion of Fisher
  consistency is more suitable and elegant for functionals than the
  usual consistency or  asymptotic unbiasedness.


 A: Older statistics books used "invariant" in a slightly different way than one might expect; the ambiguous terminology persists.  A more modern equivalent is "equivariant" (see the references at the end of this post).  In the present context it means
$$T_n(X_1+c,X_2+c,\ldots,X_n+c) = T_n(X_1,X_2,\ldots,X_n) + c$$
for all real $c$.
To address the question, then, suppose that $T_n$ has the property that for sufficiently large $n$, all real $c$, and all $m \le \varepsilon^{*}n$,
$$|T_n(\mathbf{X + Y}) - T_n(\mathbf{X})| = o(|c|)$$
whenever $\mathbf Y$ differs from $\mathbf{X}$ by at most $c$ in at most $m$ coordinates.
(This is a weaker condition than assumed in the definition of breakdown bound.  In fact, all we really need to assume is that when $n$ is sufficiently large, the expression "$o(|c|)$" is some value guaranteed to be less than $|c|/2$ in size.)
The proof is by contradiction.  Assume, accordingly, that this $T_n$ is also equivariant and suppose $\varepsilon^{*} \gt 1/2$.  Then for sufficiently large $n$, $m(n) = \lfloor \varepsilon^{*}n\rfloor$ is an integer for which both $m(n)/n \le \varepsilon^{*}$ and $(n-m(n))/n \le \varepsilon^{*}$.  For any real numbers $a,b$ define
$$t_n(a, b) = T_n(a, a, \ldots, a,\ b, b, \ldots, b)$$
where there are $m(n)$ $a$'s and $n-m(n)$ $b$'s. By changing $m(n)$ or fewer of the coordinates we conclude both
$$|t(a,b) - t(0,b)| = o(|a|)$$
and
$$|t(a,b) - t(a,0)| = o(|b|).$$
For $c\gt 0$ the triangle inequality asserts
$$\eqalign{
c = |t_n(c, c) - t_n(0, 0)| &\le |t_n(c, c) - t_n(c, 0)| + |t_n(c, 0) - t_n(0,0)| \\&= o(c) + o(c) \\&\lt c/2 + c/2 \\ &= c}$$
The strict inequality on the penultimate line is assured for sufficiently large $n$.  The contradiction it implies, $c \lt c$, proves $\varepsilon^{*} \le 1/2.$

References
E. L. Lehmann, Theory of Point Estimation. John Wiley 1983.
In the text (chapter 3, section 1) and an accompanying footnote  Lehmann writes

An estimator satisfying $\delta(X_1+a, \ldots, X_n+a) = \delta(X_1,\ldots,X_n)+a$ for all $a$ will be called equivariant ...
Some authors call such estimators "invariant."  Since this suggests that the estimator remains unchanged under $X_i^\prime = X_i+a$, it seems preferable to reserve that term for functions satisfying $u(x+a)=u(x)$ for all $x,a$.

