When Total Variation is Additive

F. S. Cater
1982 Proceedings of the American Mathematical Society  
Let / and g be continuous functions of bounded variation on [0,1]. We use the Dini dérivâtes of/and g to give a necessary and sufficient condition that the equation V(f + g) = V(f) + V(g) holds. Let/and g be continuous functions of bounded variation on [0,1]. We know that V(f+ g) < V(f) + V(g) where V denotes total variation on [0,1]. Equality holds in some special cases, for example when / and g are both nondecreasing or both nonincreasing. On the other hand, equality does not hold when / is
more » ... nconstant and g = -/. In this note we give a necessary and sufficient condition for equality to hold in terms of the four Dini dérivâtes, D+ , D+ , D~ , Z>_ , of/and g. Definition 1. We say that /is increasing (or decreasing) at a point x E (0,1) if
doi:10.2307/2044024 fatcat:evq64pq6urgxzgnj24bdazn7ci