= Line criterion for a monotone operator
{title2=$t\mapsto\langle A(u+tv),v\rangle\text{ is nondecreasing}$}
A map $A:X\to X^*$ 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 $(t-s)(\varphi(t)-\varphi(s))\geq0$. Conversely take $u=y$, $v=x-y$ and compare the endpoints to recover $\langle A(x)-A(y),x-y\rangle\geq0$. No continuity or differentiation of $A$ is needed.
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