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This post will be essentially about functions of bounded variation of one variable. The main source is the book “Functions of Bounded variation and Free Discontinuity Problems” by Ambrosio, Fusco and Pallara. Before we give the definition of a bounded variation function let us recall what exactly does is mean for a function to belong in W^{1,1}(0,1). Recall that any function f\in L^{1}(0,1) can be seen as a distribution T_{f} i.e.  as a bounded linear functional on C_{c}^{\infty}(0,1)T_{f}:C_{c}^{\infty}(0,1)\to\mathbb{R}  with

T_{f}(\phi)=\int_{0}^{1}f(x)\phi(x)dx,\quad \forall \phi\in C_{c}^{\infty}(0,1).

In that case we say that the distribution T_{f} is representable by the function f. Given any distribution T we can define its distributional derivative  DT  to be the distribution defined as

DT(\phi)=-D(\phi'),\quad \forall \phi\in C_{c}^{\infty}(0,1).

In the special case where the distribution T can be represented by a function in the way we show above the distributional derivative DT_{f} will be

DT_{f}(\phi)=-\int_{0}^{1}f(x)\phi'(x)dx,\quad \forall \phi\in C_{c}^{\infty}(0,1).

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