# Definition: Riemann Sum With Respect to a Partition

Let $$[a,b]$$ be a closed real interval, $$f:[a,b]\mapsto\mathbb R$$ be a bounded real function, and $$a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b$$ be a partition of the interval $$[a,b]$$. Further, let $$\xi\in[x_{k-1},x_k]$$ (i.e. $$\xi$$ is any point in the interval $$[x_{k-1},x_k]$$, $$1\le k\le n$$). We call

$\sum_{k=1}^n f(\xi_k)(x_k-x_{k-1})$

a Riemann sum of $$f$$ with respect to the partition $$a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b$$.

Please note that we can have different Riemann sums of $$f$$ with respect to the same partition, since the choice of the supporting points $$\xi_k$$ is arbitrary in each interval $$[x_{k-1},x_k]$$.

The mesh $$\mu$$ of the partition $$a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b$$ is defined as the maximum length of a single partition interval

$\mu:=\max_{1\le k\le n}(x_k-x_{k-1}).$

Propositions: 1

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983