A deterministic incomplete automaton A = < Q, Sigma, delta > is partially monotonic if its state set Q admits a linear order such that each partial transformation delta(_, a) with a is an element of delta preserves the restriction of the order to the domain of the transformation. We show that if A possesses a 'killer' word w is an element of Sigma(*) whose action is nowhere defined, then A is 'killed' by a word of length vertical bar Q vertical bar + [(vertical bar Q vertical bar - 1)/(2)].