It is not the system that is ill-conditioned or pathological, nor is it modelling error, I agree. It is the comparison x(t) >= h that is ill-conditioned, and yo

It is not the system that is ill-conditioned or pathological, nor is it modelling error, I agree. It is the comparison x(t) >= h that is ill-conditioned, and you'd be wrong to rely on its result (much like catastrophic cancellation, it is a bad idea to compare approximately equal numbers with errors in them, no matter where the errors come from—the errors could come from measurements, and the same logic would apply). We might have different ideas about what in your example is specified as given (with full precision) and what is specified approximately? But I do not think I misunderstood anything you wrote, please consider re-reading my comments too.