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. $\square$