We have that dual of $f$ is given by $f^*(x) = \sup_w [w^T \alpha - f(w)]$.
We want to find the dual of $f$, which is given by
$ f(z)= \begin{cases} 1 - z^Tx - \frac{\gamma}{2} &\text{if}\, 1-z^Tx > \gamma\\ \frac{1}{2\gamma}(1 - z^Tx) &\text{if}\, 1-z^Tx \in [0, \gamma]\\ 0&\text{if } 1-z^Tx < 0 \end{cases} $
I tried to consider $\nabla f(z) = \begin{cases} -x &\text{if}\, 1-z^Tx > \gamma\\ (x)\frac{1}{\gamma}(z^Tx-1) &\text{if}\, 1-z^Tx \in [0, \gamma]\\ 0&\text{if } 1-z^Tx < 0 \end{cases}$
but I'm not sure how to proceed.