The following theorem holds for BV functions (a reference for the proof is given in a question on MathOverflow.
There exists a constant $\kappa>0$ such that for every $u\in BV(\mathbb R^N)$ and $\lambda>0$ there exists a Lipschitz function $v_\lambda \in \mathrm{Lip}(\mathbb R^N)$ with Lipschitz constant $L= \lambda \kappa$ such that $$ |x\in \mathbb R^N : u(x)\ne v_\lambda(x)| \le \frac{\kappa}{\lambda} |Du|(\mathbb R^N). $$
In the case $N=1$, how can we give a simple proof of the statement?
In $1$-dimension the statement reads
There exists a constant $\kappa>0$ such that for every $u\in BV([a,b])$ and $\lambda>0$ there exists a Lipschitz function $v_\lambda \in \mathrm{Lip}([a,b])$ with Lipschitz constant $L= \lambda \kappa$ such that $$ |x\in \mathbb [a,b] : u(x)\ne v_\lambda(x)| \le \frac{\kappa}{\lambda} T.V._{[a,b]}(u). $$