1.1 The Cauchy Criterion for Integrability

Recall the definition of the (Riemann) integral:

Definition 1.1: (Riemann Integral)
Let \(f:[a,b] \to \mathbb{R}\) be bounded. Then \(f\) is Riemann integrable if \[\underline{\int_a^b} f = \overline{\int_a^b} f.\] If this happens, then the (Riemann) integral of \(f\) is defined to be the common value, and given the notation \(\int_{a}^b f.\)

Note that for a function to be integrable, we require both the upper and lower Riemann integrals to exist and be equal. These were defined as follows:

Definition 1.2: (Lower and Upper Riemann Integrals)

Let \(f:[a,b] \to \mathbb{R}\) be bounded. Then:

  • The lower Riemann integral is \[\underline{\int_a^b} f := \sup\left\lbrace L(f,P) \bigg\lvert\; \text{$P$ is a subdivision of $[a.b]$}\right\rbrace.\]
  • The upper Riemann integral is \[\overline{\int_a^b} f := \inf\left\lbrace U(f,P) \bigg\lvert\; \text{$P$ is a subdivision of $[a.b]$}\right\rbrace.\]

To actually find these values, we need to consider every possible subdivision \(P\) of the domain \([a.b]\). Doing this practically is near impossible, except in very rare cases1. What we would really like is a way of determining integrability from only a selection of partitions. It shouldn’t come as a surprise by now that such a method exists, and it’s due to — you guessed it — Cauchy!2

Proposition 1.1: (Cauchy Criterion for Integrability)

Let \(f:[a,b] \to \mathbb{R}\) be a bounded function. Then \(f\) is Riemann integrable if and only if for all \(\epsilon > 0\), there exists a subdivision \(P\) of \([a,b]\) such that \(U(f,P) - L(f,P) < \epsilon.\)

So, why is this formulation useful? Due to Archimedes’ principle, we now only have to consider regularly spaced subdivisions \(P_n\) of \([a,b]\) to determine integrability! In particular, these subdivisions are given by \[P_n = \lbrace x_0, \ldots, x_n\rbrace, \quad x_i = a + \frac{i(b-a)}{n}.\] This criterion also gives us the following theorem:

Theorem 1.2:

Let \(f:[a,b] \to \mathbb{R}\). Then

  • If \(f\) is monotonic, then it is integrable.
  • If \(f\) is continuous, then it is integrable.

So, using the Cauchy criterion, we have determined that a large class of functions are integrable! However, to prove the second part of this theorem, we require a (slightly) stronger version of continuity.

  1. Such as the function \(f\) being constant, for example.↩︎

  2. Result number six on the ‘named after Cauchy’ counter!↩︎