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When Total Variation is Additive

1982
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Proceedings of the American Mathematical Society
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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

doi:10.2307/2044024
fatcat:evq64pq6urgxzgnj24bdazn7ci