8.1. Pointwise Convergence

Example 8.1.8 (c): Pointwise Convergence does not preserve Integrability

Find a pointwise convergent sequence of Riemann-integrable functions whose limit is not Riemann-integrable.

Let rn be the (countable) set of rational numbers inside the interval [0, 1], ordered in some way, and define the functions



  • The sequence gn converges pointwise to g
  • Each gn has only finitely many points of discontinuity, thus each gn is Riemann integrable
  • The limit function g is not Riemann integrable

Thus, pointwise convergence does not preserve Riemann-integrability.

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