Integrable Functions

students, in this lesson you will learn what it means for a function to be Riemann integrable and why this idea matters in Real Analysis 📘. The main goal is to understand when a function has a well-defined area under its graph over an interval, and how upper sums, lower sums, and partitions help us decide that.

By the end of this lesson, you should be able to:

explain the meaning of an integrable function,
use the language of upper sums and lower sums,
connect integrability to Riemann Integration I,
recognize examples of functions that are integrable or not integrable,
and summarize the key ideas with clear mathematical reasoning.

A big idea to keep in mind is this: in Riemann integration, we do not start by assuming every function has an integral. Instead, we test whether the function behaves nicely enough on an interval $[a,b]$ to allow a single number to represent the total signed area. That number is the Riemann integral ✨.

What Does It Mean to Be Integrable?

Suppose $f$ is a bounded function on a closed interval $[a,b]$. To measure the area under its graph, we divide the interval into smaller pieces using a partition $P = \{x_0,x_1,\dots,x_n\}$, where $a = x_0 < x_1 < \cdots < x_n = b$.

For each subinterval $[x_{i-1},x_i]$, we look at:

the smallest values of $f$ on that piece, giving the lower sum $L(f,P)$,
the largest values of $f$ on that piece, giving the upper sum $U(f,P)$.

These sums give a range where the true area might lie. If we use finer and finer partitions, a function is integrable when the gap between upper and lower sums can be made as small as we want. In symbols, $f$ 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 means the upper and lower approximations can be squeezed together until they match up in the limit.

Another common way to say this is that the upper integral and lower integral are equal. When that happens, the common value is written as

$$\int_a^b f(x)\,dx.$$

So, students, integrability is about whether the function’s approximate areas agree from above and below 🎯.

Why Upper and Lower Sums Matter

Upper and lower sums are the core tools in deciding integrability. Think about estimating the area of a mountain range from above and below. If your estimate from above and your estimate from below keep getting closer as you use more detailed measurements, then you can trust one final number.

For a subinterval $[x_{i-1},x_i]$, let

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

Then

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

and

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

The lower sum uses rectangle heights chosen from the bottom of the graph on each subinterval. The upper sum uses rectangle heights chosen from the top. If $f$ is continuous on $[a,b]$, then the graph has no sudden jumps, and the gap between these sums becomes small for sufficiently fine partitions. That is why continuous functions are integrable.

Example: let $f(x) = x^2$ on $[0,1]$. This function is continuous, so it is integrable. If we split $[0,1]$ into many small intervals, the upper and lower rectangles nearly cover the same area. The exact integral is

$$\int_0^1 x^2\,dx = \frac{1}{3}.$$

This number represents the exact area under the curve from $0$ to $1$ 📏.

Important Facts About Integrable Functions

There are several standard facts about integrable functions in Real Analysis.

1. Every continuous function on a closed interval is integrable

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

This is one of the most important results in the topic. Continuity prevents wild behavior on the interval, so upper and lower sums can be made arbitrarily close.

2. Boundedness is necessary

A function must be bounded on $[a,b]$ to be Riemann integrable. If a function grows without bound on the interval, then the upper sums may not even be finite, so the integral cannot be defined in the usual Riemann sense.

For example, $f(x) = \frac{1}{x}$ on $[0,1]$ is not bounded, so it is not Riemann integrable on the whole interval $[0,1]$.

3. Not every bounded function is integrable

A bounded function can still fail to be integrable if it behaves too irregularly. A classic example is the Dirichlet function on $[0,1]$:

$$f(x) = \begin{cases}

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

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

$\end{cases}$$$

This function is bounded, but on every interval it takes both values $0$ and $1$. So every lower sum is $0$, and every upper sum is the length of the interval. Since the gap never shrinks, this function is not integrable.

4. Functions with only finitely many discontinuities are integrable

A bounded function on $[a,b]$ with only finitely many discontinuities is Riemann integrable. This includes many piecewise-defined functions used in applications.

For example, a step function that jumps at one or two points is still integrable, because those jump points are isolated and do not ruin the whole interval.

Examples and Non-Examples

Let us compare several functions to sharpen the idea.

Example 1: A polynomial

The function $f(x) = x^3 - 2x + 1$ is continuous everywhere, so it is integrable on every closed interval $[a,b]$. Its graph may curve up and down, but there are no breaks or jumps.

Example 2: A piecewise function

Consider

$$f(x) = \begin{cases}

1, & 0 \le x < $\frac{1}{2}$,\\

3, & $\frac{1}{2}$ \le x \le 1.

$\end{cases}$$$

This function is bounded and has just one jump at $x = \frac{1}{2}$. It is integrable on $[0,1]$. In fact, the integral is

$$\int_0^1 f(x)\,dx = \frac{1}{2}(1) + \frac{1}{2}(3) = 2.$$

This matches the area of two rectangles with widths $\frac{1}{2}$ and heights $1$ and $3$.

Example 3: The greatest integer function

The greatest integer function $f(x) = \lfloor x \rfloor$ is not continuous at integers, but on any closed interval it has only finitely many discontinuities if the interval is bounded. Therefore, it is integrable on intervals like $[0,2]$.

Non-example: Infinite oscillation near a point

Some bounded functions can fail to be integrable if they oscillate too wildly. A function that switches between many values on every neighborhood of a point may create upper and lower sums that never close the gap. The key question is not just whether the function is bounded, but whether the discontinuities are tame enough to allow approximation by rectangles.

How Integrable Functions Fit Into Riemann Integration I

Integrable functions are the final target of the Riemann integration process. The syllabus topics fit together like this:

Partitions break the interval into smaller parts.
Upper and lower sums estimate area from above and below.
Integrability asks whether those estimates can be made arbitrarily close.
Integrable functions are exactly the functions for which this works.

So integrable functions are not a separate topic floating on its own. They are the natural conclusion of the whole method of Riemann Integration I. Without partitions, there are no sums. Without sums, there is no test for integrability. Without integrability, there is no Riemann integral.

This is why the concept is so important in Real Analysis: it explains which functions can be measured using the Riemann approach and which cannot. It also prepares you for deeper ideas later, such as more general integration methods and stronger convergence theorems.

Conclusion

students, the main idea of integrable functions is simple but powerful: a bounded function on $[a,b]$ is Riemann integrable when its upper and lower sum approximations can be forced together so tightly that they define one unique number. That number is the integral.

Continuous functions are integrable, many piecewise functions are integrable, and functions with only finitely many discontinuities are integrable. But functions that are unbounded or too wildly discontinuous may fail to be integrable.

Understanding integrable functions helps you see the purpose of partitions, upper sums, and lower sums. It turns Riemann Integration I from a set of formulas into a clear idea about approximating area with increasing precision 📚.

Study Notes

An integrable function is a bounded function on $[a,b]$ whose upper and lower sums can be made arbitrarily close.
The formal test is: for every $\varepsilon > 0$, there exists a partition $P$ such that $U(f,P) - L(f,P) < \varepsilon$.
If $f$ is continuous on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.
A bounded function with only finitely many discontinuities is integrable.
A function must be bounded to be Riemann integrable on a closed interval.
The Dirichlet function is bounded but not integrable on $[0,1]$ because it oscillates between $0$ and $1$ on every interval.
Upper sums use supremum values and lower sums use infimum values on each subinterval.
Integrability is the bridge between approximating area and computing the exact value of $\int_a^b f(x)\,dx$.
In Riemann Integration I, integrable functions are the functions for which the process works successfully.