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**manjohn12** Here is the definition: Let $\displaystyle a,b \in \mathbb{R} $ with $\displaystyle a<b $. Let $\displaystyle f $ be a real valued function whose domain consists of the interval $\displaystyle [a,b] $. We say that $\displaystyle f $ is Riemann integrable on $\displaystyle [a,b] $ if there exists a real number $\displaystyle I $ such that for all $\displaystyle \epsilon >0 $, there exists $\displaystyle \delta >0 $ such that $\displaystyle |\mathcal{R}(f,P)-I| < \epsilon $ whenever $\displaystyle \mathcal{R}(f,P) $ is a Riemann sum for $\displaystyle f $ corresponding to a partition of $\displaystyle [a,b] $ with mesh less than $\displaystyle \delta $.