ThmDex – An index of mathematical definitions, results, and conjectures.
Subconvex real function need not have a minimizer
Formulation 0
Let $f : \mathbb{R} \to \mathbb{R}$ be a D4364: Real function such that
(i) \begin{equation} f(x) = x \end{equation}
Then
(1) $f$ is a D5606: Subconvex real function
(2) \begin{equation} \underset{x \in \mathbb{R}}{\text{arg min}} \, f(x) = \emptyset \end{equation}
Proofs
Proof 0
Let $f : \mathbb{R} \to \mathbb{R}$ be a D4364: Real function such that
(i) \begin{equation} f(x) = x \end{equation}
Clear. $\square$