Bounded variation functions and some properties on metric spaces

Abstract

Bounded variation functions of a single variable were first introduced by Camille Jordan (1881). Bounded variation functions or \U0001d435\U0001d449 functions is a function with the total variation is finite. The variation in a function aims to measure how much increase and decline occure in the function. In this paper, we generalize this function in the metric space. Let (\U0001d44b, \U0001d451) be a complete metric space and \U0001d6fe: \U0001d446 → ℝ is a function, where \U0001d446 is a closed and bounded subset of \U0001d44b. The function \U0001d6fe is of bounded variation on \U0001d446 if \U0001d449(\U0001d6fe, \U0001d446) is finite, i.e \U0001d449(\U0001d6fe, \U0001d446) < ∞. The result of this paper is the definition and properties algebraic of bounded variation function on metric spaces. Some properties of this function is multiplication with scalar, sum of two functions, and product of two functions. The function of bounded variation to \U0001d446 are also bounded variation to each subspace from \U0001d446. If \U0001d6fe is a bounded variation function on \U0001d446 then \U0001d6fe is bounded on \U0001d446. We also consider the related topic it is absolute continuity. If a function \U0001d6fe is absolutely continuous on \U0001d446 then \U0001d6fe is of bounded variation on \U0001d446.