 Argmax

The argmax of a function is the set of inputs where its largest value is attained: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.