Alphabeta Math

The Riemann Integral: Definition and Integrability: Examples and Counterexamples

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1 · Prerequisites
Abstract Algebra
Number Theory
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

∫₀¹ x² = 1/3, computed from the Darboux definition with uniform partitions and the closed form ∑_k<n k² = n(n-1)(2n-1)/6

Example

One refinement worked out for f(x) = x² on [0,1]: adding the point 1/2 to the trivial partition raises the lower sum from 0 to 1/8 and lowers the upper sum from 1 to 5/8

Example

∫₀³ ⌊ x ⌋ = 3: the floor function is nondecreasing, hence integrable, and the integral is computed from the uniform partitions

Example

The indicator of the Cantor set is discontinuous exactly on the Cantor set, which is null, so it is Riemann integrable with integral 0 even though it is discontinuous at uncountably many points

Example

Thomae's function is Riemann integrable on [0,1] with integral 0: it is continuous at every irrational, so its discontinuity set is countable, and every lower Darboux sum is 0

Counterexample

The Dirichlet function on [0,1] has lower Darboux integral 0 and upper Darboux integral 1, so it is bounded and not Riemann integrable

Counterexample

The indicator of the Smith-Volterra-Cantor set is discontinuous exactly on a nowhere dense set, and is not Riemann integrable, because that set does not have measure zero

Example

For every F_σ subset E of [0,1] of measure zero there is a bounded Riemann integrable function on [0,1] whose set of discontinuities is exactly E

Counterexample

For the Dirichlet function every uniform partition with rational tags gives Riemann sum 1, so the sums converge along that sequence of tagged partitions although the function is not integrable: the mesh condition of the Riemann definition quantifies over all tagged partitions and cannot be weakened to one sequence

Counterexample

Thomae's function is nonnegative, Riemann integrable on [0,1] with integral 0, and nonzero at every rational, so a vanishing integral does not force a nonnegative integrand to vanish

Sources