Is there any function∇ f which is monotone but f does not exist or is not convex i.e counter example to [∇ f is monotone ⟹ f is convex ]?

Question

I tried making such piecewise functions but didn't work.There are answers (How to show that the rotation map $f$ is not a gradient of a convex function?)saying that the rotation matrix which is skew symmetric is such an operator as the hessian should be symmetric but hessian and gradient are not the same isn't it?

Answer 1

If $f \colon \mathbb R^n \to \mathbb R$ is differentiable, then $f$ is convex if and only if $\nabla f$ is (maximal) monotone.

However, there are (maximal) monotone operators $A \colon \mathbb R^n \to \mathbb R^n$, which are not the gradient of a convex function.

Indeed, any skew symmetric $A$ works (here, we identify $A$ with the operator $x \mapsto A \, x$). If there would be a differentiable function $f \colon \mathbb R^n \to \mathbb R$ with $\nabla f(x) = A\, x$, then $f$ would be twice differentiable with $\nabla^2 f(x) = A$. However, this contradicts symmetry of the Hessian.