Riemann Integrable Functions

Real Analysis · Axiom Academy

LESSON Riemann Integrable Functions Which bounded functions can be integrated — the upper/lower-sum criterion, the classes that always pass it, and the famous one that fails. 1. The Integrability Criterion Partition [a,b] into subintervals. On each one, the tallest the function reaches gives a rectangle in the upper sum U(P,f) ; the shortest gives one in the lower sum L(P,f) . Always . The function is integrable when refining the partition drives the gap U(P,f)-L(P,f) to zero. 2. Continuous Functions Are Integrable A continuous function on a closed interval is uniformly continuous: one works everywhere. So if , every piece varies by less than , and the gap between upper and lower sums is at most times the total width. Why it works. By uniform continuity, choose so that . With the sup and inf on each piece differ by less than , so For f(x)=x^2 on [0,1] the gap is exactly 1/n with n equal pieces: . The shrinking column in the animation tracks that same gap on a wavier curve. 3. Monotone Functions Are Integrable A monotone function needn't be continuous — it can jump. But monotonicity forces something powerful: on each piece the sup is the right endpoint and the inf is the left endpoint, so the per-piece gaps telescope . Slide them together and they stack into a single column. The telescoping bound. With a uniform mesh , summing the right-minus-left gaps collapses to the total rise of the function: 4. The Dirichlet Function Is Not Integrable

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