For an M-estimator defined by $\sum_i \psi(X_i - \hat\theta) = 0$, the influence function at $x$ is $IF(x; T, F) = \psi(x - \theta_0) / \int \psi'(y - \theta_0)\, dF(y)$. Why does a *bounded* $\psi$ matter for robustness?

Robust Statistics and M-Estimators

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For an M-estimator defined by \sum_{i} ψ (X_{i} - \hat{θ}) = 0, the influence function at x is I F (x; T, F) = ψ (x - θ_{0}) / \int ψ^{'} (y - θ_{0}) d F (y) . Why does a *bounded* ψ matter for robustness?