Upper sum

An upper sum of a bounded function $f:[a,b]\to\mathbb R$ with respect to a partition $P=\{x_0,\dots,x_n\}$ is the number

where $M_i=\sup\{f(x):x\in[x_{i-1},x_i]\}$ is the supremum of $f$ on the $i$th subinterval.

Upper sums, together with lower sums , give the standard criterion for a Riemann integrable function : upper and lower sums can be made arbitrarily close by choosing a suitable partition.

Examples:

• For $f(x)=x$ on $[0,1]$ and $P=\{0,\tfrac12,1\}$, one gets $U(f,P)=\tfrac12\cdot\tfrac12+1\cdot\tfrac12=\tfrac34$.
• If $f(x)=c$ is constant on $[a,b]$, then $U(f,P)=c(b-a)$ for every partition $P$.