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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