1. It decreases by vanishingly smaller amount.
2. It decreases to negative infinity.
So...which is it? Infinitesimal time units, or negative time?
Rounding the decrease to the nearest time unit some of the time is hardly a disqualifier for "decreasing". (Though specifically they said "faster".)
The mathematically rigorous form of the posed question is whether there is an algorithm with run time f(N) for which there exists strictly decreasing g(N) such that f(N) = O(g(N)). That is, the run time is bounded by a forever decreasing function.
f(N) itself doesn't need to be strictly decreasing, but there needs to be a g(N) that is.
Bottom line, no algorithm can have a forever decreasing run time as N increases. An algorithm can exhibit that behavior over a limited range of N, but that's not how complexity works.
Let's say my minimum runtime is 7, and the strictly decreasing bound on my runtime is g(N) = 1e15/N + 7
Doesn't that fit your description fine?
Assume that you use a linkedlist for collisions, then for some growth rates, your cost of access is not constant, and becomes linear to your input.
Actually, it is impossible to construct an algorithm with this property, at least under any usual computational model.
Every computational model has discrete time units; therefore the amount of time taken cannot be vanishingly small.
(This is assuming the usual notion of complexity, where only the asymptotic behavior matters. It doesn't matter if it gets increasingly faster over the first million input sizes...it only matters what the behavior is as n -> infinity.)
At the limit, the functional cannot be increasingly faster.