Convex conjugate of a differentiable function

Question

Let $f : \mathbb{R}^n \to \mathbb{R}$ be convex and differentiable everywhere. For $y \in \mathbb{R}^n$, define $$f^*(y) := \sup\limits_{x \in \mathbb{R}^n} \lbrace y\cdot x - f(x)\rbrace$$ Define $D = \lbrace\nabla f(x): x \in \mathbb{R}^n\rbrace$. Is it the case that if $y \not\in D$, then $f^*(y) = \infty$?

Answer 1

It is false. Set $n=1$ and take $f(x) = e^x$. Then $0\not\in D$ but $f^*(0) = \sup_{x\in\mathbb{R}}\lbrace-e^x\rbrace=0$.