Multiplicity function for functions of bounded variation
Kelleher, Brother Roch
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Considerable study h8s been devoted to the multiplicity function of a real variable. For any real valued function of a real variable, f(x), define its multiplicity function, N(Y), as the cardinal number of roots, either finite or infinite of y = f(x), i.e., when the cardinal number of roots is transfinite, assign the value infinite where it is understood that the range of the multiplicity function is the extended real number system. Banach^1 was the first to relate this function to a continuous function of bounded variation. He realized that the integral of this function over the Entire real line was precisely the total variation. He demonstrated analogous theorems for curves and surfaces. His approach is to express the curve on the surface parametrically. In the case of a simple arc in the plane, the original theorem can be applied to the parametric equations. This results in an expression for rectifiable arcs. To determine a surface of finite area requires a more careful study. Here, too, the method of parametric equations simplifies the problem. The proof consists chiefly in defining and in organizing the expressions of the surface in a manner that will allow the multiplicity theorem to be applied [TRUNCATED].
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