How equality in Fenchel-Young inequality characterizes subdifferential?

We will show that $$f(x)+f^*(u) = \langle x,u \rangle \Longleftrightarrow u \in \partial f(x).$$

Indeed, for any proper, lsc, convex function f, we have \begin{align*} u \in \partial f(x) &\Longleftrightarrow f(z) \geq f(x) + \langle u, z-x\rangle \quad\forall z \\ &\Longleftrightarrow \langle u, x\rangle - f(x) \ge \langle u, z\rangle - f(z) \quad\forall z \\ &\Longleftrightarrow \langle u, x\rangle - f(x) = \sup_z\left\{ \langle u, z\rangle - f(z)\right\} \\ &\Longleftrightarrow \langle u, x\rangle - f(x) = f^*(u) \\ &\Longleftrightarrow f(x)+f^*(u) = \langle x,u \rangle, \text{QED}. \end{align*}

See Rockafellar et al 2009, Variational analysis, proposition 11.3 for details.