= Essential smoothness of a convex function
= Essential smoothness
{synonym}
= Essentially smooth
{synonym}
An extended-real <convex function> $\Lambda:\mathbb R^d\to\mathbb R\cup\{+\infty\}$ is <essentially smooth> if its effective domain has nonempty interior, it is differentiable throughout that interior, and $\|\nabla\Lambda(\theta_n)\|\to\infty$ whenever interior points converge to a finite boundary point of the domain. A differentiable <convex function> finite on all of $\mathbb R^d$ satisfies the boundary condition vacuously. The <cumulant-generating function> $\log(r/(r-\theta))$ on $\theta<r$ has derivative $(r-\theta)^{-1}$ and is <essentially smooth>. Restricting its domain to $\theta<\lambda<r$ destroys this property. This distinction matters in the lower bounds of the <Gärtner–Ellis theorem>.
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