Line criterion for a monotone operator ($t\mapsto ⟨A(u+tv),v⟩\text{ is nondecreasing}$)

A map $A:X\to X^{\ast }$ on a real Banach space is a monotone operator exactly when the displayed scalar function is nondecreasing on $[0,1]$ for every $u,v\in X$. Applying operator monotonicity to $u+tv$ and $u+sv$, $s<t$, gives $(t-s)(\phi (t)-\phi (s))\ge 0$. Conversely take $u=y$, $v=x-y$ and compare the endpoints to recover $⟨A(x)-A(y),x-y⟩\ge 0$. No continuity or differentiation of $A$ is needed.