Real Analysis II · Integration
Two-jump integrator
Problem
Fix 0<c<d<1 and let α(x)=0 for x<c, α(x)=A for c≤x<d, and α(x)=A+B for d≤x≤1. Prove that ∫_0^1 f dα=A f(c)+B f(d) for every continuous f.
Hint
Insert c and d into the partition; refinement does not change the candidate limit.
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