Tag condition for Riemann integrability

Question

As part of the definition for a definite integral we say that the choice of tag in the partition should not affect the limit of the Riemann sum for that function to be considered Riemann integrable. What is an example of a function where the location of the tag changes the limit of the sum?

Answer 1

I will assume that you mean the choice of intermediate points associated with a given partition $P.$ Consider the function $\chi := \chi_{\mathbb{Q} \cap [0, 1]},$ the indicator function of the rationals in $[0, 1].$ Let us choose a partition $P := (x_0 = 0 < x_1 < \dots < x_{n - 1} < x_n = 1)$ of the interval $[0, 1]$ and choose two systems of intermediate points $\xi := (\xi_1, \dots, \xi_n),$ $\zeta := (\zeta_1, \dots, \zeta_n),$ where $\xi_j, \zeta_j \in [x_{j - 1}, x_j]$ for each $1 \leq j \leq n$ such that $\xi_j \in \mathbb{Q}$ and $\zeta_j \not\in \mathbb{Q}$ for all $1 \leq j \leq n.$ Then $\sum_{1 \leq j} \chi(\xi_j)(x_j - x_{j - 1}) = \sum_{1 \leq j \leq n} (x_j - x_{j - 1}) = 1$ and $\sum_{1 \leq j \leq n} \chi(\zeta_j) (x_j - x_{j - 1}) = 0,$ so this function is not Riemann integrable because it matters which system of intermediate points you choose to compute the Riemann sum.

The behaviour of this function under Riemann integrability is even nastier than this in fact. You can easily show that you may find some partitions and certain systems of intermediate points such that the associated Riemann sum converges to any number you like $\lambda \in [0, 1].$ However, this function is Lebesgue integrable and its integral is equal to $0.$ Hope this helps. :)