Giving 2 intervals $I,J\in\mathbb{R}$ and 2 continuous functions $u: I\rightarrow\mathbb{R}^d$ and $v: J\rightarrow \mathbb{R}^d$, we say that $u$ and $v$ are equivalent and we write $u\approx v$ if there exists an increasing function (possibly not strictly increasing) $\phi : I\rightarrow J$ such that $\phi(I)=J$, and for every discontinuity point $x\in I$ of $\phi$, the interval $[\phi_{-}(x),\phi_{+}(x)]$ is contained in an interval $[\tau_1,\tau_2]\subset J$ on which $v(y)=c(\tau_1,\tau_2)\in\mathbb{R}^d$ for all $y\in [\tau_1,\tau_2]$, and $u(x)=v(\phi(x))$ for all $x\in I$.
Prove that $\approx$ is an equavalent relation. Can any one help me with this problem ?