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In what ways is the influence function a different measure of resistance for an estimator?

In what ways is the influence function a different measure of resistance for an estimator?

Math   Statistics-And-Probability   
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The influence function actually measres the amount of infinitesimal change of the estimator (or the functional) when a single point contamination is added to the dataset. If it is bounded for all point contaminations, it means whatever be the amount of contamination (that is model misspecification), the estimator can not be driven a lot from the true value of the parameter. In this way the influence function is a measure of resistance/robustness. And since it deals with the infinitesimal change of the estimator, it is different from the asymptotic breakdown point measure of robustness.

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