Clinical trials are monitored periodically for safety, efficacy, and futility. The spending function is a popular tool for efficacy monitoring because it does not require pre-specification of the number or timing of interim analyses. But there are infinitely many spending functions, so some guidance on how to choose one is helpful. We consider spending functions that are generated from different continuous-monitoring boundaries for Brownian motion. We use properties of the continuous-monitoring boundaries to derive properties of the associated spending function. Copyright © 2012 John Wiley & Sons, Ltd.