Given two intervals $I=[a,b],J=[c,d]\subseteq\mathbb{R}$ and two continuous functions $\boldsymbol{u}:I\rightarrow\mathbb{R}^{d}$ and $\boldsymbol{v}% :J\rightarrow\mathbb{R}^{d}$, we say that $\boldsymbol{u}$ and $\boldsymbol{v}% $ are equivalent, and we write $\boldsymbol{u}\approx\boldsymbol{v}$, if there exists an increasing function (possibly not strictly increasing) $\phi:I\rightarrow J$ such that the smallest interval that contains $\phi\left( I\right) $ is $J$, for every discontinuity point of $x\in I$ of $\phi$, the interval $\left[ \phi_{-}\left( x\right) ,\phi_{+}\left( x\right) \right] $ is contained in an interval $\left[ \tau_{1},\tau _{2}\right] \subseteq J$ on which $\boldsymbol{v}$ is constant, and $\boldsymbol{u}(x)=\boldsymbol{v}(\phi(x))$ for all $x\in I$.
Prove that $\approx$ is an equivalent relation. Can anyone help me with that ?