Question

Define Riemann integrability in two different ways and prove their equivalence.
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Answer 1

Answer:

we say that the riemann integral of f equals s if the following condition holds: ... for a given ε>0, there exists δ such that for any tagged partition x0,⋯,xn and t0,⋯,tn−1 whose mesh is less than δ, we have |n−1∑i=0f(ti)(xi+1−xi)−s|<ε.

Answer 2

Answer:

We say that the Riemann integral of f equals s if the following condition holds: ... For a given ε>0, there exists δ such that for any tagged partition x0,⋯,xn and t0,⋯,tn−1 whose mesh is less than δ, we have |n−1∑i=0f(ti)(xi+1−xi)−s|<ε.