What is Influence function, breakdown point?

For an estimator written as a functional $T(F)$ of the data’s law, the influence function at $x$ is $\mathrm{IF}(x) = \lim_{\varepsilon \to 0}\bigl(T((1 - \varepsilon)F + \varepsilon\delta_x) - T(F)\bigr)/\varepsilon$: the effect on the estimate of an infinitesimal contamination at the point $x$ (Hampel, 1974). The breakdown point is the smallest fraction of the sample that, replaced by arbitrary values, can carry the estimate arbitrarily far.