    \begin{align*}
        g_c &= c ( g(u) - c_r g_{r}(u) - c_l g_{l}(u))  \\
        &= c \cdot \begin{cases}
            \frac{h_1}{\beta} u \hspace{40mm} &u \in [0, \beta] \\
            h_2 + \frac{h_3 - h_2 }{\alpha- \beta}(u-\beta) - h_2 + h_1
            & u \in (\beta, \alpha) \\
            1 - (1- h_3) - (h_2 - h_1) & u \in [\alpha,1] 
        \end{cases} \\
        &= c \cdot \begin{cases}
            \frac{h_1}{\beta} u \hspace{40mm} & u \in [0, \beta] \\
            h_1 + \frac{h_3 - h_2}{\alpha - \beta} (u - \beta) 
            & u \in ( \beta, \alpha) \\
            h_3 - h_2 + h_1  & u \in [\alpha,1]
        \end{cases}.
    \end{align*}
