Darboux Sums and Integrals

Vocabulary flashcards covering Darboux sums, subinterval bounds, and upper and lower Darboux integrals based on the mathematical analysis lecture notes.

Partition of an interval $[a, b]$

A finite set of points $P = \{x_k\}_{k=0}^n$ dividing an interval $[a, b]$ such that $a = x_0 < x_1 < \dots < x_n = b$.

Subinterval Value Set $E_k$

The set of all values taken by a function $f$ on the subinterval $[x_{k-1}, x_k]$.

Subinterval Supremum $M_k$

The supremum of the set $E_k$ of function values on the subinterval $[x_{k-1}, x_k]$, defined as $M_k = \sup E_k$.

Subinterval Infimum $m_k$

The infimum of the set $E_k$ of function values on the subinterval $[x_{k-1}, x_k]$, defined as $m_k = \inf E_k$.

Upper Darboux Sum $U(f, P)$

The sum defined as $U(f, P) = \sum_{k=1}^n M_k (x_k - x_{k-1})$, representing the upper bound area constructed over partition $P$.

Lower Darboux Sum $L(f, P)$

The sum defined as $L(f, P) = \sum_{k=1}^n m_k (x_k - x_{k-1})$, representing the lower bound area constructed over partition $P$.

Upper Darboux Integral $I^*(f)$

The infimum of the upper Darboux sums over all possible partitions $P$ of the interval, defined as $I^*(f) = \inf_P U(f, P)$.

Lower Darboux Integral $I_*(f)$

The supremum of the lower Darboux sums over all possible partitions $P$ of the interval, defined as $I_*(f) = \sup_P L(f, P)$.