This question is related to Convex conjugate of a differentiable function
Let $f : \mathbb{R}^n \to \mathbb{R}$ be convex and differentiable everywhere. For $y \in \mathbb{R}^n$, define $$f^*(y) := \sup\limits_{x \in \mathbb{R}^n} \lbrace y\cdot x - f(x)\rbrace.$$ Set $D = \lbrace\nabla f(x): x \in \mathbb{R}^n\rbrace$. Let $\overline{\mathrm{conv}D}$ be the closed convex hull of $D$. Is it the case that if $y \not\in \overline{\mathrm{conv}D}$, then $f^*(y) = \infty$?