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.