II.1.2.4: Alternative Definitions

Suppose $f$ and $g$ are arbitrary real-valued functions on $\mathcal{X}$. Define $$\begin{align*} \mathcal{S}_-(f,g):=&\{x\in\mathcal{X}\mid f(x)\leq g(x)\},\\ \mathcal{S}_0(f,g):=&\{x\in\mathcal{X}\mid f(x)=g(x)\}. \end{align*}$$ Thus $\mathcal{S}_0(f,g)\subseteq\mathcal{S}_-(f,g).$ Then if $$\begin{align*} x&\in\mathcal{S}_0(f,g),\\ y&\in\mathop{\mathbf{argmin}}_{x\in\mathcal{S}_-(f,g)}g(x), \end{align*}$$ we have $$f(y)\leq g(y)\leq g(x)=f(x).$$