Some remarks on functions of Λ-bounded variation

Abstract

It is shown that if a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda BV"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>BV</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Lambda {\text {BV}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> function has no external saltus, then its total <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -variation is independent of its values at points of discontinuity, and a function which is equal to the given function at points of continuity cannot have a lesser total <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -variation. Necessary and sufficient conditions are determined for one <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda BV"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>BV</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Lambda {\text {BV}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space to contain another and for two spaces to be identical.