Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $f \in \mathcal{C}^{1}$ with real values. We assume that the set $C(f)$ of points in $]-\pi, \pi[$ where the function $f^{\prime}$ vanishes is finite. We denote by $\ell$ the cardinality of $C(f)$ and, if $\ell \geq 1$, we denote by $t_{1} < \cdots < t_{\ell}$ the elements of $C(f)$. We set $t_{0} = -\pi$ and $t_{\ell+1} = \pi$.
Show that $$V(f) = \sum_{j=0}^{\ell} \left|f\left(t_{j+1}\right) - f\left(t_{j}\right)\right|$$
For $0 \leq j \leq \ell$, let $\psi_{j}$ be the function from $\mathbf{R}$ to $\{0,1\}$ equal to 1 on $\left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$ and to 0 on $\mathbf{R} \backslash \left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$. Show that $$V(f) = \sum_{j=0}^{\ell} \int_{-\|f\|_{\infty}}^{\|f\|_{\infty}} \psi_{j}$$