This paper arose from the need for a better understanding of the minimal-order observer problem. A deadbeat function observer is an estimator that estimates exactly a linear function of the state of a deterministic discrete-time linear system via an incomplete state observation. The minimal order of a minimal-time function observer, which exhibits the fastest deadbeat response among deadbeat function observers, was obtained and such an observer was designed. It was shown that the minimal order of a minimal-time function observer is not necessarily the same as the minimal order of a deadbeat function observer.