# Total variation of a function

1. Dec 12, 2011

### brian_m.

Hello,

I want to calculate the total variation $\left \| f \right \|_{V(\Omega)}$ with $\Omega=(-1,1)$ and $f(x)=\mathrm{sgn}(x)$.

The total variation of a function is defined as follows:

$\left \| f \right \|_{V(\Omega)} :=\sup\left \{ \int_\Omega f\ \mathrm{div} (v)\ dx \ | \ v \in C^1_c(\Omega)^n \text{ with } \left \| v \right \|_{\infty,\Omega}\leq 1 \right \}$

So, this is a very abstract definition and I don't know how to apply it...