A function's robustness is determined by the component with the worst asymptotic growth rate

This is very like the Big-O notation in a sense that the worst component dominates, while others go invisible. Instead of computing performance, we compute the robustness against extreme values. The influence function measures the impact of a single observation on the function’s output as the observation grows extreme.

see The Influence Curve and its Role in Robust Estimation for the research paper defining the influence function see Robust outlier detection requires robustness on both location and dispersion layers for an instance of that in practice (EDA) see Avoiding outlier magnitudes makes the measure robust for a complementary information on primitive function components