We consider a marked empirical process corresponding to a sample X₁, ..., X_n of independent and identically distributed random variables and exchangeable random marks C₁, ..., C_n. If the sum of all marks is equal to zero, then there exists a certain order statistic with random index, which is a maximizing point of the process. It is shown that for each finite sample size n ∈ N this point of maximum has the same distribution as X₁ (argmax-stability). As an application we derive the exact distribution of an estimator for the discontinuity point in a regression function.