Upper Sum and Lower Sum of a Function on a Partition

definitionAnalysis

Upper Sum and Lower Sum of a Function on a Partition

definitionAnalysisdef:upper-lower-sums-partition-c54-2026a
· by ChatGPT-5.4, Aaron ·
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Reason: Publish upper and lower sum definition used in integrability arguments.

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}, and let

P:a=x0<x1<⋯<xn=bP:a=x_0<x_1<\cdots<x_n=b

be a \reftext{def:partition-closed-interval-c54-2026a}{partition} of [a,b][a,b]. For each i=1,…,ni=1,\dots,n, let

Mi=sup⁡{f(t):t∈[xi−1,xi]},mi=\reftextdef:lower−bound−infimum−c54−2026ainf⁡{f(t):t∈[xi−1,xi]}.M_i=\sup\{f(t): t\in[x_{i-1},x_i]\},\qquad m_i=\reftext{def:lower-bound-infimum-c54-2026a}{\inf}\{f(t): t\in[x_{i-1},x_i]\}.

The upper sum and lower sum of ff with respect to PP are defined by

U(f,P)=∑i=1nMi(xi−xi−1),L(f,P)=∑i=1nmi(xi−xi−1).U(f,P)=\sum_{i=1}^n M_i(x_i-x_{i-1}),\qquad L(f,P)=\sum_{i=1}^n m_i(x_i-x_{i-1}).
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