Integrability in Riemann Integration I

students, imagine trying to measure the area of an oddly shaped garden using only rectangles 🌱. You can’t always use a simple formula, so you make better and better rectangle estimates. That idea is the heart of Riemann integration, and integrability tells us when those rectangle estimates actually settle down to one exact number.

In this lesson, you will learn how integrability works, why it matters, and how it connects partitions, upper and lower sums, and the broader picture of Real Analysis.

What Integrability Means

The main goal of Riemann integration is to define the area under a curve in a way that is mathematically precise. For a bounded function $f$ on an interval $[a,b]$, we divide the interval into smaller pieces using a partition

$$P = \{x_0, x_1, \dots, x_n\}$$

where

$$a = x_0 < x_1 < \cdots < x_n = b.$$

On each subinterval $[x_{i-1}, x_i]$, we look at how large and how small the function gets. Define

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

and

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

Then the upper sum and lower sum are

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

and

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

These sums are like two estimates of the same area: the upper sum overestimates, and the lower sum underestimates. If the function is well-behaved enough, these two estimates can be made as close as we want.

A bounded function $f$ is Riemann integrable on $[a,b]$ if the upper and lower sums can be made arbitrarily close. A common formal way to say this is that for every $\varepsilon > 0$, there exists a partition $P$ such that

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

That is the key idea of integrability: the gap between the top estimate and the bottom estimate can be squeezed to zero.

Why Upper and Lower Sums Matter

Upper and lower sums are important because they turn the vague idea of “area” into something precise. students, think about pricing a carpet for a room with an uneven outline 🏠. If you estimate using rectangles that stick above the actual outline, you get a larger cost. If you estimate using rectangles that stay below it, you get a smaller cost. If both estimates can be made to agree as closely as needed, then the true cost is well defined.

For a function graph, the same logic applies. The upper sum captures the tallest possible rectangle on each subinterval, while the lower sum captures the shortest possible one. Refining the partition usually makes these estimates tighter.

If a function has a lot of wild behavior, the gap between $U(f,P)$ and $L(f,P)$ may never disappear. In that case, the function is not Riemann integrable. So integrability is not just about having an area-like quantity; it is about whether the function can be measured accurately by rectangles.

A useful fact is that if a function is continuous on $[a,b]$, then it is Riemann integrable on $[a,b]$. This is one of the most important examples in Real Analysis. Smooth graphs, polynomial graphs, and many familiar functions are integrable because their behavior is stable enough on small intervals.

The Intuition Behind Integrability

Integrability is really about controlling error. When you use one big rectangle to estimate a curve, the error might be large. When you use many thin rectangles, the error usually gets smaller.

Imagine the graph of $f(x) = x^2$ on $[0,1]$. The function is continuous and smooth. If we split $[0,1]$ into many small subintervals, then on each piece the function barely changes. That means the upper and lower sums get very close. Since the difference between these sums can be made smaller than any chosen $\varepsilon > 0$, the function is integrable.

Now compare that with a function that jumps abruptly, such as a step function. Some step functions are still integrable because the jump happens on only a few points or on a small set of points. But if a function is too irregular everywhere, it may fail to be integrable. The key question is not whether a function looks rough, but whether the roughness prevents the upper and lower estimates from converging together.

This is why Real Analysis pays so much attention to the behavior of functions on intervals. Integrability depends on how the function behaves over tiny pieces, not just on its overall shape.

Examples and Non-Examples

Let’s look at some examples.

Example 1: A continuous function

Take $f(x) = x^2$ on $[0,1]$. Since $f$ is continuous on a closed interval, it is Riemann integrable. If we choose a partition with many equally spaced points, the maximum and minimum values of $f$ on each subinterval become very close. As the partition gets finer, the upper and lower sums approach the same limit.

This limit is the Riemann integral:

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

The integral gives the exact area under the curve between $x=0$ and $x=1$.

Example 2: A step function

Define a function by

$$f(x) = \begin{cases} 1, & 0 \le x < \frac{1}{2} \\ 2, & \frac{1}{2} \le x \le 1. \end{cases}$$

This function jumps at $x = \frac{1}{2}$. But it is still Riemann integrable because the jump is limited to one point. A partition that includes $x = \frac{1}{2}$ makes the upper and lower sums easy to analyze. In fact,

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

Even though the function is not continuous everywhere, it is still integrable.

Example 3: A non-integrable function

Consider the Dirichlet function on $[0,1]$:

$$f(x) = \begin{cases} 1, & x \in \mathbb{Q} \\ 0, & x \notin \mathbb{Q}. \end{cases}$$

Every interval, no matter how small, contains both rational and irrational numbers. So on every subinterval, the supremum is $1$ and the infimum is $0$. That means for any partition $P$,

$$U(f,P) = 1$$

and

$$L(f,P) = 0.$$

So the difference is always

$$U(f,P) - L(f,P) = 1,$$

which can never be made smaller than an arbitrary $\varepsilon > 0$. Therefore, this function is not Riemann integrable on $[0,1]$.

This example shows the core test for integrability: if the function oscillates too much on every interval, the upper and lower sums do not converge together.

How Integrability Fits Into Riemann Integration I

Integrability is the bridge between approximation and exactness. In Riemann Integration I, students first learn how partitions cut an interval into pieces and how upper and lower sums create estimates. Integrability is the next step: it tells us when those estimates define a real number.

So the sequence of ideas is:

1. Start with a bounded function on $[a,b]$.
2. Choose a partition $P$.
3. Build upper and lower sums $U(f,P)$ and $L(f,P)$.
4. Check whether the difference can be made arbitrarily small.
5. If yes, the function is integrable, and the common limit is the Riemann integral.

This matters because integration is used to measure many things in mathematics and science: area, accumulated change, total distance from velocity, average value, and more. But all of these depend on the function being integrable.

A powerful theorem in Real Analysis says that if $f$ is continuous on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$. Another important idea is that bounded functions with only finitely many discontinuities are also Riemann integrable. These results show that integrability is quite broad, but not universal.

Concluding View of Integrability

Integrability answers a simple but deep question: can a function be measured accurately by rectangles? 📏 If the answer is yes, then the function has a Riemann integral. If the answer is no, then the function’s behavior is too irregular for this method.

In Real Analysis, this lesson is important because it connects the geometry of areas, the logic of limits, and the precision of function behavior. students, when you understand integrability, you understand why some functions can be integrated and others cannot, and you see how partitions and sums create the foundation of Riemann integration.

Study Notes

• A partition of $[a,b]$ is a finite set of points $a = x_0 < x_1 < \cdots < x_n = b$.
• On each subinterval $[x_{i-1}, x_i]$, define $M_i$ using the supremum and $m_i$ using the infimum.
• The upper sum is $U(f,P) = \sum_{i=1}^n M_i(x_i - x_{i-1})$.
• The lower sum is $L(f,P) = \sum_{i=1}^n m_i(x_i - x_{i-1})$.
• A bounded function $f$ is Riemann integrable on $[a,b]$ if for every $\varepsilon > 0$, some partition $P$ satisfies $U(f,P) - L(f,P) < \varepsilon$.
• Integrability means the upper and lower estimates can be made arbitrarily close.
• Continuous functions on $[a,b]$ are Riemann integrable.
• Step functions with finitely many jumps are Riemann integrable.
• The Dirichlet function on $[0,1]$ is not Riemann integrable because every interval contains both rational and irrational numbers.
• Integrability is the foundation that allows the Riemann integral to be defined as a precise number.