Additive and bounded functions of curves
Maria, Alfred Joseph
Master of Arts
In the present paper we apply the analysis of Vitali to functions of curves of limited variation* To determine the structure of functions of limited variation we associate with every finite function an additive function, called a discard. This discard measures, in a true manner, the quantity by which the function ceases to be absolutely continuous. We demonstrate that the property which characterizes a discard is that the discard coincide with its own discard. We prove the theorem that every function of limited variation can be decomposed into the sum of a function of point values, a continuous function and a finite or infinite number of elementary discards each multiplied by a constant. Finally use is made of the preceding results to find the structure of a function of limited variation hut having no point values.