Partition and Darboux sums

For a bounded function \(f\) on \([a,b]\), a partition is a list \(a=x_0<x_1<\dots<x_n=b\). On each subinterval \([x_{i-1},x_i]\) one takes \(m_i=\inf f\) and \(M_i=\sup f\), which exist by completeness of the reals; \(m_i\Delta x_i\) is the area of an inscribed rectangle and \(M_i\Delta x_i\) that of a circumscribed one. When \(f\) is continuous, the infimum and supremum are actually a minimum and a maximum; otherwise they exist only as bounds.

Guiding example: \(f(x)=x\) on \([0,1]\) with \(P=\{0,\tfrac12,1\}\) gives \(L=\tfrac14\) and \(U=\tfrac34\), with the area \(\tfrac12\) in between. If \(f\) increases, \(m_i\) sits at the left endpoint and \(M_i\) at the right; if it decreases, the reverse. A common mistake is to always use left and right endpoints without looking at how \(f\) behaves.

Always \(L(f,P)\le U(f,P)\). Moreover, a refinement \(Q\supseteq P\) tightens both sums: \(L(f,P)\le L(f,Q)\le U(f,Q)\le U(f,P)\). Merging two partitions into their common refinement shows that every lower sum is at most every upper sum, even if they come from different partitions.

That last property is what allows taking the supremum of the lower sums and the infimum of the upper sums, which is exactly what the Riemann integral does. For the Dirichlet function, by contrast, \(L=0\) and \(U=1\) for every partition: the sums never get closer.