Instantaneous measures

I've been reading about Influence functions,

$${d}/{d ε}(t((1-ε)F+ε I_{[y,∞)}))_{ε=0}$$.

These are used to quantify the influence a given data point has on a statistic $t$.

I was thinking about this as an instantaneous rate in the same sense as a hazard function.

In the limiting notation we can see how they are both types of averages across an increment and then the increment is decreased to approach 0 from above to give the derivative in that direction at a point.

So the influence function is an average difference between the statistic of interest between the 2 distributions (one being a mixture distribution). The hazard rate is an average probability of transitioning within the time interval.

The influence function is slightly different to the hazard rate because it includes a wieghted sum of cdfs to sum to 1 in the functional.