MONOTON FUNKSIYANING UZLUKSIZLIGI ISBOTI
Keywords:
Kalit so’zlar:Monotonlik, uzluksizlik, limit, chap limit, o‘ng limit, tartibli to‘plam, to‘liqlik aksiomasi, sakrash uzilishlari, Lebeg o‘lchovi., Keywords: Monotonicity, continuity, limit, left limit, right limit, ordered set, completeness axiom, jump discontinuities, first kind discontinuity, almost universal continuity, Lebesgue measuAbstract
Ushbu ilmiy maqolada monoton funksiyalarning analitik xossalari, xususan ularning uzluksizligi masalasi keng qamrovda tadqiq etilgan. Monotonlikning matematik tabiati, chap va o‘ng limitlarning mavjudligi, haqiqiy sonlar to‘plamining to‘liqligi va tartib xossalari asosida monoton funksiyaning uzluksizligi teoremasi bosqichma-bosqich isbotlangan. Shuningdek, sakrash nuqtalarining tuzilishi, ularning sanoqliligi va Lebeg o‘lchovi bo‘yicha nolga tengligi ko‘rsatib berilgan.
Annotation:This scientific article comprehensively studies the analytical properties of monotonic functions, in particular the issue of their continuity. The mathematical nature of monotonicity, the existence of left and right limits, the completeness of the set of real numbers, and the continuity theorem of a monotonic function are proven step by step based on the order properties. The structure of jump points, their countability, and their equality to zero in terms of Lebesgue measure are also shown. The article is enriched with theoretical analyses, examples, practical applications, graphical interpretations, and other mathematical models. The results are significant in their application in mathematical analysis, functional analysis, economic models, optimization, and other areas.
