Solution
= Solution
The functional is
$$
T(F)=\int\psi(x)\,dF(x),
\qquad
\psi(x)=\Delta\left(x-\frac{\theta_0+\theta_1}{2}\right).
$$
The <influence function> of an expectation functional is
$$
\operatorname{IF}_{\rm test}(x;T,F_\theta)
=\psi(x)-\mathbb E_{F_\theta}\psi(X).
$$
Because $\psi$ is affine with nonzero slope, this is unbounded under both $F_{\theta_0}$ and $F_{\theta_1}$.