Riemann Integration: Sums, Integrability, and Core Properties

All flashcards

Riemann integrability

A bounded function is Riemann integrable if all sufficiently fine tagged sums approach one value, uniformly over tag choices.

Riemann sum

For a tagged partition, the sum of each function value at its tag multiplied by its subinterval width.

Darboux criterion

A bounded function is Riemann integrable if upper and lower sums can be made arbitrarily close.

Uniform-limit theorem for Riemann integrals

A uniform limit of Riemann-integrable functions on a closed bounded interval is integrable, and its integral is the limit of their integrals.

Improper integral

An integral over an unbounded interval defined as a limit of integrals over bounded intervals.

Quiz questions

1. For a bounded function on a closed interval to be Riemann integrable with value $s$, what must hold for every $\varepsilon>0$?Answer: Some mesh bound makes every tagged sum differ from $s$ by less than $\varepsilon$.
2. On $[0,1]$, let $I_{\mathbb Q}(x)=1$ for rational $x$ and $0$ for irrational $x$. Why is this function not Riemann integrable?Answer: Every subinterval has infimum $0$ and supremum $1$, so the upper-minus-lower sum gap remains $1$.
3. What condition characterizes Riemann integrability for a bounded function on a compact interval?Answer: Its set of discontinuities has measure zero.
4. If Riemann-integrable functions $f_n$ converge uniformly to $f$ on $[a,b]$, which conclusion holds?Answer: $f$ is Riemann integrable and its integral is the limit of the integrals of $f_n$.
5. For $\operatorname{sgn}(x)$ on the whole real line, symmetric integrals over $[-a,a]$ equal zero. What does this imply about its improper integral over $\mathbb R$?Answer: The improper integral does not exist, although its Cauchy principal value is zero.