Integrability Criteria in Riemann Integration II

students, in this lesson you will learn how to decide when a bounded function is Riemann integrable and why those tests matter in Real Analysis 📘. The big idea is simple: not every function with values on an interval can be integrated using Riemann sums, but many important ones can. By the end, you should be able to explain the main integrability criteria, use them to test functions, and connect them to the bigger picture of Riemann Integration II.

Why integrability criteria matter

The Riemann integral is built from rectangles. To find the area under a curve over $[a,b]$, we split the interval into small pieces, choose sample points, and add up rectangle areas. If the upper sums and lower sums get closer and closer as the partition gets finer, the function is Riemann integrable.

But how do we know whether that happens? That is where integrability criteria come in. They give practical and theoretical ways to decide whether a function is integrable without checking every possible partition by hand.

A key point is that for a bounded function $f$ on $[a,b]$, Riemann integrability is not automatic. For example, the function

$$

$ f(x)=$

$ \begin{cases}$

$ 1, & x\in \mathbb{Q}, \\$

$ 0, & x\notin \mathbb{Q}$

$ \end{cases}$

$$

on $[0,1]$ is bounded, but it is not Riemann integrable because every interval contains both rational and irrational numbers, so its upper sum is always $1$ and its lower sum is always $0$. This example shows why a criterion is useful: it separates functions that behave nicely from those that do not.

The basic definition behind the criteria

To understand the criteria, students, it helps to recall the core definition. Let $P$ be a partition of $[a,b]$:

$$

$P=\{x_0,x_1,\dots,x_n\}, \quad a=x_0<x_1<\cdots<x_n=b.$

$$

For each subinterval $[x_{i-1},x_i]$, define

$$

$M_i=\sup\{f(x):x\in[x_{i-1},x_i]\}, \qquad m_i=\inf\{f(x):x\in[x_{i-1},x_i]\}.$

$$

The upper and lower sums are

$$

U(f,P)=$\sum_{i=1}$^n M_i(x_i-x_{i-1}), \qquad L(f,P)=$\sum_{i=1}$^n m_i(x_i-x_{i-1}).

$$

A bounded function $f$ is Riemann integrable on $[a,b]$ if for every $\varepsilon>0$, there exists a partition $P$ such that

$$

$U(f,P)-L(f,P)<\varepsilon.$

$$

This is the main definition behind the integrability criteria. The criteria are different ways of proving or recognizing this condition.

The Lebesgue criterion for Riemann integrability

The most important integrability criterion in this topic is the Lebesgue criterion. It says:

A bounded function $f$ on $[a,b]$ is Riemann integrable if and only if the set of its points of discontinuity has measure zero.

In simpler language, students, this means the function may fail to be continuous at some points, but those bad points must be rare enough. A set has measure zero if, roughly speaking, it can be covered by intervals whose total length is as small as we want.

This criterion explains many familiar examples:

• Every continuous function on $[a,b]$ is Riemann integrable, because it has no discontinuities.
• A function with only finitely many discontinuities is Riemann integrable.
• A function with countably many discontinuities can still be Riemann integrable if those discontinuities form a set of measure zero.

For example, the step function

$$

$ f(x)=$

$ \begin{cases}$

$ 0, & x<\tfrac12, \\$

$ 1, & x\ge \tfrac12$

$ \end{cases}$

$$

is Riemann integrable on $[0,1]$. It has only one discontinuity, at $x=\tfrac12$, so the discontinuity set has measure zero.

Easy integrability tests you can use

There are several useful tests that follow from the main criterion.

1. Continuous functions are integrable

If $f$ is continuous on a closed interval $[a,b]$, then $f$ is Riemann integrable.

Why? A continuous function on a closed interval is bounded and has no discontinuities. So the discontinuity set is empty, and the Lebesgue criterion applies.

This is one of the most important facts in the course, because many functions in calculus are continuous, such as polynomials, trigonometric functions, and exponential functions.

2. Functions with finitely many discontinuities are integrable

If a bounded function has only finitely many discontinuities on $[a,b]$, then it is Riemann integrable.

A bounded function with finitely many “bad points” still behaves well enough because a finite set has measure zero. The discontinuities can be isolated in tiny intervals whose total length is made arbitrarily small.

3. Monotone functions are integrable

If $f$ is monotone on $[a,b]$, then $f$ is Riemann integrable.

A monotone function may have jump discontinuities, but the set of discontinuities of a monotone function is at most countable, hence measure zero. So monotone behavior is enough for integrability.

For example, $f(x)=\lfloor x \rfloor$ on $[0,3]$ is monotone and has jumps only at integers. Therefore it is Riemann integrable.

4. Piecewise continuous functions are integrable

If a function is continuous on each piece of a finite partition of $[a,b]$ and bounded at the breakpoints, then it is Riemann integrable.

This is very common in applications. Signals, graphs in engineering, and many “real world” functions are built from smooth pieces joined together at a few corners or jumps. Those corner points do not stop integrability.

How to apply an integrability criterion

When students sees a new function, a good strategy is to ask these questions:

1. Is the function bounded on the interval?
2. Is it continuous everywhere on the interval?
3. If not, where are the discontinuities?
4. Are there only finitely many discontinuities?
5. If there are infinitely many discontinuities, do they form a set of measure zero?

Let’s do two examples.

Example 1: A polynomial

Consider $f(x)=x^2-4x+1$ on $[0,5]$.

Since polynomials are continuous everywhere, $f$ is continuous on $[0,5]$. Therefore $f$ is Riemann integrable.

This example is simple, but it shows the power of the criterion: no sum calculations are needed to prove integrability.

Example 2: A function with rational and irrational behavior

Define

$$

$ f(x)=$

$ \begin{cases}$

$ x, & x\in \mathbb{Q}, \\$

$ 0, & x\notin \mathbb{Q}$

$ \end{cases}$

$$

on $[0,1]$.

This function is bounded, but it is discontinuous at every point of $[0,1]$. Why? In every neighborhood of any point, there are both rational and irrational numbers, so the function values jump around too much. The set of discontinuities is all of $[0,1]$, which does not have measure zero. So this function is not Riemann integrable.

This example is important because it shows that boundedness alone is not enough.

Connection to upper and lower sums

Integrability criteria are not separate from the definition; they are ways of understanding it. If a function has only a small set of discontinuities, then most subintervals in a fine partition contain values of $f$ that do not vary much. That makes the difference $U(f,P)-L(f,P)$ small.

For a continuous function, uniform continuity on $[a,b]$ guarantees that over sufficiently small intervals, the oscillation of $f$ is small. That means upper and lower sums can be forced close together.

For a function with a few discontinuities, you can choose a partition that isolates the bad points in tiny intervals. Outside those intervals, the function behaves nicely, and the rest of the interval contributes little to the gap between upper and lower sums.

This is the real mechanism behind many integrability proofs: control the bad part, and show the good part is tame.

Conclusion

Integrability criteria are the practical tools that tell students whether a bounded function can be integrated using Riemann sums. The most important result is the Lebesgue criterion: a bounded function on $[a,b]$ is Riemann integrable exactly when its discontinuities form a set of measure zero. From this, we get major tests such as “continuous implies integrable,” “finitely many discontinuities imply integrable,” and “monotone implies integrable.” These ideas fit directly into Riemann Integration II and prepare you for deeper results like the Fundamental Theorem of Calculus, where integrable functions play a central role. ✅

Study Notes

• A bounded function $f$ on $[a,b]$ is Riemann integrable if for every $\varepsilon>0$, some partition $P$ satisfies $U(f,P)-L(f,P)<\varepsilon$.
• The Lebesgue criterion says that $f$ is Riemann integrable if and only if the set of discontinuities of $f$ has measure zero.
• Every continuous function on $[a,b]$ is Riemann integrable.
• A bounded function with finitely many discontinuities is Riemann integrable.
• Every monotone function on $[a,b]$ is Riemann integrable.
• Piecewise continuous functions on a closed interval are Riemann integrable.
• The function that equals $1$ on $\mathbb{Q}$ and $0$ on $\mathbb{R}\setminus\mathbb{Q}$ is not Riemann integrable on $[0,1]$.
• To test a function, check boundedness, continuity, and the size of the discontinuity set.
• Integrability criteria explain why upper and lower sums can be made arbitrarily close.
• These ideas are a foundation for the rest of Riemann Integration II and later results in Real Analysis.