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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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