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pasmith
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There is a more elegant proof.
Notation-wise, you will confuse the reader and yourself if you have x_i being both the points at which f(x) \neq g(x) and the points of an arbitrary partition. So I will call the former \{\xi_k : k = 1, \dots, n\}.
We can write f(x) = g(x) + \sum_{k=1}^n (f(\xi_k) - g(\xi_k)) h_k(x) where <br /> h_k : [a,b] \to \mathbb{R} : x \mapsto \begin{cases} 0 & x \neq \xi_k \\ 1 & x = \xi_k.\end{cases} It suffices to show that each h_k is (Darboux) integrable, since a linear combination of finitely many (Darboux) integrable functions is (Darboux) integrable. (The one with the upper and lower sums being arbitrarily close to each other is Darboux integrability; it is however completely equivalent to Riemann integrability.)
Let \epsilon > 0 and D = \{x_0 = a, \dots, x_m = b\} a partition of [a.b] with \|D\| = \max\{\delta_i = x_i - x_{i-1}, i = 1, \dots, m\} < \frac12 \epsilon. Then there exists a minimal 1 \leq i_1\leq m such that x_{i_1-1} \leq \xi_k \leq x_{i_1}. We then have that M_i - m_i = 0 for every i except for i_1 and, if x_{i_1} = \xi_k and i_1 < m, also i_1+1. For these intervals M_i - m_i = 1. Thus (M_i - m_i)\delta_i = 0 except for at most two intervals for which (M_i - m_i)\delta_i = \delta_i \leq \|D\|. Thus <br /> U(h_k,D) - L(h_k,D) \leq 2\|D\| < \epsilon.
Notation-wise, you will confuse the reader and yourself if you have x_i being both the points at which f(x) \neq g(x) and the points of an arbitrary partition. So I will call the former \{\xi_k : k = 1, \dots, n\}.
We can write f(x) = g(x) + \sum_{k=1}^n (f(\xi_k) - g(\xi_k)) h_k(x) where <br /> h_k : [a,b] \to \mathbb{R} : x \mapsto \begin{cases} 0 & x \neq \xi_k \\ 1 & x = \xi_k.\end{cases} It suffices to show that each h_k is (Darboux) integrable, since a linear combination of finitely many (Darboux) integrable functions is (Darboux) integrable. (The one with the upper and lower sums being arbitrarily close to each other is Darboux integrability; it is however completely equivalent to Riemann integrability.)
Let \epsilon > 0 and D = \{x_0 = a, \dots, x_m = b\} a partition of [a.b] with \|D\| = \max\{\delta_i = x_i - x_{i-1}, i = 1, \dots, m\} < \frac12 \epsilon. Then there exists a minimal 1 \leq i_1\leq m such that x_{i_1-1} \leq \xi_k \leq x_{i_1}. We then have that M_i - m_i = 0 for every i except for i_1 and, if x_{i_1} = \xi_k and i_1 < m, also i_1+1. For these intervals M_i - m_i = 1. Thus (M_i - m_i)\delta_i = 0 except for at most two intervals for which (M_i - m_i)\delta_i = \delta_i \leq \|D\|. Thus <br /> U(h_k,D) - L(h_k,D) \leq 2\|D\| < \epsilon.