The proof relied on a detailed analysis of subdifferentiation for convex, nonsmooth functions.

The research paper examined the various subdifferentiations of the tumor to improve targeted therapies.

The subdifferentiating operator provided a robust way to analyze nonsmooth functions.

When solving nonsmooth optimization problems, the subdifferentials at boundary points can be nonempty but fail to be singletons.

When analyzing convex functions, the subdifferential at a point can provide a set of supporting slopes that generalize the derivative.

In convex analysis, the function is subdifferentiable at the boundary point even though it is not differentiable there.

The convex function was analyzed subdifferentially to determine the stability of its minima.