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Sufficient conditions for integrability

This post is about single variable functions $f : [a, b] \to \mathbb{R}$ defined in a finite closed interval, and about some sufficient conditions for their integrability in the interval $[a, b]$. Here $a$ and $b$ are real numbers of course.

Integrability refers to the existence of the definite integral 

$$\int\limits_a^b f(x)\,dx$$

One should note that there are different definitions of integrability (i.e. different definitions of the definite integral shown above).

The one I am referring to here is Darboux integrability. This one is equivalent to Riemann integrability. 

Then there is also Lebesgue integrability which allows a wider family of functions to be classified as integrable. So all functions which are Darboux/Riemann integrable are also Lebesgue integrable but the converse is not true. This is a very interesting topic and relates to measure theory.

OK... so here are several sufficient conditions for Darboux/Riemann integrability. 

Theorem 1: If the function $f : [a, b] \to \mathbb{R}$ is defined and continuous in the finite closed interval $[a,b]$, it is integrable in $[a,b]$. 

Theorem 2: If the function $f : [a, b] \to \mathbb{R}$ is defined and bounded in the finite closed interval $[a,b]$, and if it is discontinuous only at a finite number of points, then it is integrable in $[a,b]$. 

Theorem 3: If the function $f : [a, b] \to \mathbb{R}$ is defined and monotonic in the finite closed interval $[a,b]$, then it is integrable in $[a,b]$.

Note that the last theorem does not disallow a situation where $f$ has infinitely many points of discontinuity. That situation is allowed and the function if still integrable (provided that it's monotonic, as the theorem says).

Again, it should be pointed out that these are just sufficient conditions for integrability, they are not necessary. There are functions which do not satisfy the conditions of any of these three theorems but are still integrable. 

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