Argmax (source code)
= Argmax
{title2=$\operatorname*{arg\,max}_{x\in X}f(x)$}
The <argmax> of a <function> is the <set> of inputs where its largest value is attained:
$$
\operatorname*{arg\,max}_{x\in X}f(x)=\{x\in X:f(x)\ge f(y)\text{ for every }y\in X\}.
$$
It records <global maximizers>, whereas $\max_{x\in X}f(x)$ records the value. For $f(x)=-(x-2)^2$, the <argmax> on $\mathbb R$ is $\{2\}$ and the maximum value is zero. If $f$ is constant, every input belongs to the <argmax>.
The <argmax> can be empty: $f(x)=x$ on $(0,1)$ has <supremum> one but never attains it. A <continuous function> on a nonempty <compact set> does attain its maximum by the <extreme value theorem>. Writing a single input as an <argmax> therefore implicitly assumes uniqueness or specifies a rule for choosing between ties. This distinction matters when an <optimization> problem is used to define an <estimator>.
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