The Additivity Theorem

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The Additivity Theorem for Riemann Integrable Functions: Let be a real-valued function on the interval , and let . Then, is Riemann integrable on if and only if it is also Riemann integrable on and . In this case,

  • Proof: Suppose that and for some . Let be given.
    • Since we have that for there exists a partition such that for all partitions finer than , () and for any choice of 's in each subinterval we have that:
    • Similarly, since we have that for there exists a partition such that for all partitions finer than , ) and for any choice of 's in each subinterval we have that:
    • Let . Then is a partition of and for all partitions finer than , () we must have that and hold. Then for any choice of 's in each subinterval we have that:
    • Hence exists and:


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