The Riemann Integral and When a Function Is Integrable
Define the integral by upper and lower sums and call a function integrable when the two meet. Continuous functions qualify; a function that is discontinuous everywhere need not.
What a learner can do afterwards
- Compute upper and lower sums for a partition and compare them
- State the criterion for integrability in terms of refinements
- Give a bounded function that is not Riemann integrable
1 · Read
On each piece of a partition, take the infimum of f for the lower sum and the supremum for the upper sum, multiply each by the piece's width, and add. For an increasing function these values sit at the left and right endpoints. For f(x) = x on [0, 2] with partition {0, 1, 2}, the lower sum is 0 times 1 plus 1 times 1, which is 1, and the upper sum is 1 times 1 plus 2 times 1, which is 3. The exact integral, 2, lies between them.
Refinement can never lower the lower sum or raise the upper sum. A bounded function is Riemann integrable when, for every epsilon greater than 0, some partition makes the upper sum minus the lower sum less than epsilon. A continuous function on a closed interval is uniformly continuous, so narrow pieces make the gap small. Continuity is not necessary: at a single jump, only pieces containing the jump keep a gap, and narrowing them shrinks it.
The Dirichlet function equals 1 on rationals and 0 on irrationals. On [0, 2], every subinterval contains both kinds of points, so each supremum is 1 and each infimum is 0. Upper sums stay at 2 and lower sums stay at 0 for every partition. The gap never closes, so the function is not Riemann integrable there.
Lower sums rise, upper sums fall, and integrability means refinements can make their gap as small as any positive number.
Lesson, questions and flashcards are original text written for this dataset.
The two rewritten reading notes were written for this dataset. The third reading note and all questions are original to this dataset, and flashcards were drafted from collected fact candidates.
2 · Watch
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Optional connections
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.