I have seen this in textbooks, and indeed found this Stack thread about how to create the bar in LaTEX, but don't know what it means:


  • 4
    $\begingroup$ In textbooks, they probably introduce/explain the notation...? It's likely to be an upper (Darboux) integral (and with a bar below for the lower integral); see here. $\endgroup$
    – StackTD
    May 10, 2017 at 14:23
  • 3
    $\begingroup$ Could they be Darboux integrals? $\endgroup$
    – lioness99a
    May 10, 2017 at 14:24
  • $\begingroup$ Ah thank you, turns out it was Riemann integrals which is the same thing :) $\endgroup$
    – user296950
    May 10, 2017 at 14:27
  • $\begingroup$ As a side note, sometimes $\overline{\lim} a_n$ and $\underline{\lim} a_n$ are used to denote limit superior and limit inferior respectively. $\endgroup$
    – Alex Vong
    May 10, 2017 at 16:27

1 Answer 1


These are denoted as the upper and lower Riemann integrals respectively.

More so, we can construct the following:

Let $P=\left\{x_0,x_1,...,x_n\right\}$ be a partition on $[a,b]$ for $n\in \mathbb{N}$.

Notating the lower and upper sums of some function $f$ with respect to its partition $P$, as $L(P,f)$ and $U(P,f)$, we can define the following:

$$\underline {\int_a^b}f=sup\left\{L(P,f):\forall \ partitions \ P \ on \ [a,b]\right\}$$

$$\overline {\int_a^b}f=inf\left\{U(P,f):\forall \ partitions \ P \ on \ [a,b]\right\}$$

Furthermore, we note that for a reimann integral to exist on some bounded interval $[a,b]$,

$$\underline {\int_a^b}f=\int_a^bf=\overline {\int_a^b}f$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.