Summary: We present a result applicable to classification learning algorithms that generate decision trees of rules using the information entropy minimization heuristic for discretizing continuous-valued attributes. The result serves to give a better understanding of the entropy measure, to point out that the behavior of the information entropy heuristic possesses desirable properties that justify its usage in a formal sense, and to improve the efficiency of evaluating continuous-valued attributes for cut value selection. Along with the formal proof, we present empirical results that demonstrate the theoretically expected reduction in evaluation effort for training data sets from real-world domains.