Math 2.0

statement theorem

Monotone functions are Riemann integrable

Statement

Let a<ba < b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be monotone. Then ff is Riemann integrable on [a,b][a, b].

Remarks

Assume ff increasing, and take the partition into nn equal pieces. Then Mi=f(xi)M_i = f(x_i) and mi=f(xi−1)m_i = f(x_{i-1}), so the sum telescopes: U(f,P)−L(f,P)=b−an(f(b)−f(a)).U(f, P) - L(f, P) = \frac{b - a}{n} (f(b) - f(a)). This tends to 00, so Riemann's criterion applies. A monotone function can have infinitely many discontinuities, and it is still integrable.

Explanations

L1 human

For an increasing function, the gaps between upper and lower rectangles stack up to one column of total height f(b)−f(a)f(b) - f(a). Its width shrinks to 00 as the partition gets finer.

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To read this node, read these 14 nodes in this order. The foundations come first, and the list ends with this node.

Partition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionStrictly monotone function, definition, no evidence levelStrictly monotone fun…definitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionUpper and lower Darboux sums, definition, no evidence levelUpper and lower Darbo…definitionUpper and lower Darboux integrals, definition, no evidence levelUpper and lower Darbo…definitionDarboux and Riemann integrals agree, statement, no evidence levelDarboux and Riemann i…statementRiemann's criterion for integrability, statement, no evidence levelRiemann's criterion f…statementMonotone functions are Riemann integrable, statement, no evidence levelMonotone functions ar…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  1. Partition of an interval definition
  2. Sequence definition
  3. Strictly monotone function definition
  4. Mesh of a partition definition
  5. Riemann sum definition
  6. Real sequence definition
  7. Riemann integrable function definition
  8. Bounded sequence definition
  9. Riemann integral definition
  10. Upper and lower Darboux sums definition
  11. Upper and lower Darboux integrals definition
  12. Darboux and Riemann integrals agree statement
  13. Riemann's criterion for integrability statement
  14. Monotone functions are Riemann integrable statement

Sources

  1. Web https://en.wikipedia.org/wiki/Riemann_integral, Integrability

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