I am interested in functions $\phi:\mathbb R^d\to\mathbb R^d$ with the following property. For all $x,y\in\mathbb R^d, (\phi(x)-\phi(y))^\top (x-y)\geq 0$.
For instance, if $\phi$ is the gradient (or a subgradients) of a convex function, it satisfies this property, but this property is more general.
Examples of such functions include functions of the form $\phi=f+g$ where $f$ is a subgradients of an $\alpha$-strongly convex function and $g$ is any Lipschitz function with Lipschitz constant at most $\alpha$, where $\alpha$ could be any positive number. Other examples are given by $x\mapsto Ax$ where $A$ is any $d\times d$ matrix such that $A+A^\top$ is positive semi-definite.
First, is the set of zeros necessarily convex (or empty)? Note that if the monotonicity condition that defines such a function (see above) is a strict inequality whenever $x\neq y$, then the set of zeros is either empty or a singleton. Second, are there non-trivial properties satisfied by such functions? For instance, are they always continuous everywhere outside an at most countable subset? Can they always be decomposed as the sum of a subgradient of a convex function and a simple function (in any possible sense)?