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Consider the two following definitions of Riemann integral:

1) A function $f$ is Riemann integrable if there exist a finite no. $I$ such that $\forall \epsilon > 0$ there exist a $\delta_\epsilon >0$ such that "for all" partitions $P$ with mesh(P) (length of the largest subinterval of $P$)

2) A function $f$ is Riemann integrable if there exist a finite no. $I$ such that $\forall \epsilon > 0$ "there exist" a partition $P$ such that $|R(f,P,T) - I| < \epsilon$ "for all" $t_i$ where $t_i$ lies between two consecutive points in the partition.

Are these two definitions equivalent ?

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Yes, they are equivalent. The first is Darboux's definition, while the second is Riemann's definition. In the second definition, you also have to incude a statement about the mesh of the partition.

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