Let $f$ be a closed concave function on $N_{\mathbb{R}}$. For $u\in N_{\mathbb{R}}$ and $x\in M_{\mathbb{R}}$, the following conditions are equivalent:

1. (1)

   $x\in\partial f(u)$;

2. (2)

   $u\in\partial f^{\vee}(x)$;

3. (3)

   $P_{f}(u,x)=0$.