| Summary: | We consider the problem of extending a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt"></mspace><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> defined on a subset <i> P </i> of an arbitrary set <i> X </i> to <i> X </i> strictly monotonically with respect to a preorder ≽ defined on <i> X </i>, without imposing continuity constraints. We show that whenever ≽ has a utility representation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt"></mspace><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> is extendable if and only if it is gap-safe increasing. This property means that whenever <i><b>x</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mo>≻</mo></mrow></semantics></math></inline-formula><i><b>x</b></i>, the infimum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt"></mspace><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> on the upper contour of <i><b>x</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>′</mo></msup></semantics></math></inline-formula> exceeds the supremum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> on the lower contour of <i><b>x</b></i>, where <i><b>x</b></i>, <i><b>x</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mo>∈</mo><mover accent="true"><mi>X</mi><mo>˜</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>X</mi><mo>˜</mo></mover></semantics></math></inline-formula> is <i> X </i> completed with two absolute ≽-extrema and, moreover, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> is weakly increasing. The completion of <i> X </i> makes the condition sufficient. The proposed method of extension is flexible in the sense that for any bounded utility representation <i> u </i> of ≽, it provides an extension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> that coincides with <i> u </i> on a region of <i> X </i> that includes the set of <i> P</i>-neutral elements of <i> X </i>. An analysis of related topological theorems shows that the results obtained are not their consequences. The necessary and sufficient condition of extendability and the form of the extension are simplified when <i> P </i> is a Pareto set.
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