A simple example would be a psuedorandom number generator. For example, f(t) = f(t-1)^2 + 1. Thus, if f(0) = 0 (nothing at t=0), then f(1) = 1.

The only way to get out of boundary conditions is to define the whole universe in one step. For example, f(t) = t^3 + 3*t^2 + 1, in which case you wouldn't have causality at all.