Question 8.T.8: Let c ∈ (a, b). Then f ∈ R(a, b) if and only if f ∈ R(a, c) ...
Let c ∈ (a, b). Then f ∈ \mathcal{R}(a, b) if and only if f ∈ \mathcal{R}(a, c) ∩ \mathcal{R}(c, b) , in which case
\int_{a}^{b}{f} = \int_{a}^{c}{f} + \int_{c}^{b}{f}. (8.10)
Suppose first that f ∈ \mathcal{R}(a, b) . If ε > 0 is given, we can choose a partition P = \left\{x_{0}, x_{1}, …, x_{n} \right\} of [a, b] such that
U (f, P) − L (f, P) < ε.
By refining P if needed, we may assume that c is a member of P, say c = x_{k} . It is obvious that Q = \left\{x_{0}, x_{1}, …, x_{k} \right\} is a partition of [a, c] and that Q^{\prime} = \left\{x_{k}, x_{k+1}, · · · , x_{n} \right\} is a partition [c, b]. Since M_{i} − m_{i} ≥ 0 for all i = 0, 1, · · · , n − 1,
U (f, Q) − L (f, Q) = \sum\limits_{i=0}^{k−1}{(M_{i} − m_{i}) (x_{i+1} − x_{i})}
≤ \sum\limits_{i=0}^{n−1}{(M_{i} − m_{i}) (x_{i+1} − x_{i})}
= U (f, P) − L (f, P) < ε.
This proves that f ∈ \mathcal{R}(a, c) . Similarly, using the partition Q^{\prime} = \left\{x_{k}, x_{k+1}, · · · , x_{n} \right\} , of [c, b] , we can show that f ∈ \mathcal{R}(a, c) .
On the other hand, if f ∈ \mathcal{R}(a, c) ∩ \mathcal{R}(c, b), then, given ε > 0, we can choose a partition Q = \left\{x_{0}, x_{1}, …, x_{k} \right\} of [a, c] and a partition Q^{\prime} = \left\{x_{k}, x_{k+1}, · · · , x_{n} \right\} of [c, b] such that
U (f, Q) − L (f, Q) < ε/2, U \left(f, Q^{\prime} \right) − L \left(f, Q^{\prime} \right) < ε/2.
The partition P = \left\{x_{0}, x_{1}, …, x_{n} \right\} satisfies
U (f, P ) − L (f, P) = U (f, Q) − L (f, Q) + U \left(f, Q^{\prime} \right) − L \left(f, Q^{\prime} \right) < ε,
which proves f ∈ \mathcal{R}(a, b) . Furthermore, if \left\|Q\right\| → 0 and \left\|Q^{\prime}\right\| → 0, then \left\|P\right\| → 0. Since
U (f, P) = U (f, Q) + U \left(f, Q^{\prime} \right) ,
equation (8.10) follows from Darboux’s theorem.