of globally integrable functions, then the function space defined is that of functions of locally bounded variation. Precisely, developing this idea for , a local variation is defined as follows, for every set , having defined as the set of all precompact open subsets of with respect to the standard topology of finite-dimensional vector spaces, and correspondingly the class of functions of locally bounded variation is defined as Notation There are basically two distinct conventions for the notation of spaces of functions of locally or globally bounded variation, and unfortunately they are quite similar: the first one, which is the one adopted in this entry, is used for example in references (partially), (partially), and is the following one identifies the space of functions of globally bounded variation identifies the space of functions of locally bounded variation The second one, which is adopted in references and (partially), is the following: identifies the space of functions of globally bounded variation identifies the space of functions of locally bounded variation Basic properties Only the properties common to functions of one variable and to functions of several variables will be considered in the following, and proofs will be carried on only for functions of several variables since the proof for the case of one variable is a straightforward adaptation of the several variables case: also, in each section it will be stated if the property is shared also by functions of locally bounded variation or not. References , and are extensively used. BV functions have only jump-type or removable discontinuities In the case of one variable, the assertion is clear: for each point in the interval of definition of the function , either one of the following two assertions is true while both limits exist and are finite. In the case of functions of several variables, there are some premises to understand: first of all, there is a continuum of directions along which it is possible to approach a given point belonging to the domain ⊂. It is necessary to make precise a suitable concept of limit: choosing a unit vector it is possible to divide in two sets Then for each point belonging to the domain of the BV function , only one of the following two assertions is true or belongs to a subset of having zero -dimensional Hausdorff measure. The quantities are called approximate limits of the BV function at the