Upper and Lower Integrals

Defining the area under a curve is a cornerstone of calculus, but how is this done rigorously, especially for functions that aren't simple geometric shapes? The intuitive idea of summing infinite, infinitesimally thin rectangles runs into logical hurdles. This article addresses this foundational problem by exploring the concepts of upper and lower integrals—the elegant machinery that underpins Riemann integration. By trapping the "true area" between a guaranteed over-estimate and a guaranteed under-estimate, this method provides a robust and precise definition of integrability. This article will guide you through this powerful idea. First, in "Principles and Mechanisms," we will build the theory from the ground up, defining upper and lower sums and establishing the famous Riemann criterion for when a function can be integrated. Then, in "Applications and Interdisciplinary Connections," we will see how these principles apply to a zoo of bizarre functions, revealing how the limits of Riemann integration inspired more advanced theories and connect to fields like physics and probability theory.

Imagine you are tasked with a seemingly simple problem: finding the area of a shape with a curved top. If it were a rectangle, you'd multiply base by height. But what do you do when the "height" is constantly changing, described by some function $f(x)$? The brilliant strategy, conceived by thinkers like Georg Friedrich Bernhard Riemann, is not to find the area directly, but to trap it. We'll establish an undeniable lower bound and an undeniable upper bound for the area, and then we'll squeeze them together until they meet. This very "squeeze" is the heart of integration, and its mechanics are revealed through the beautiful concepts of ​​lower and upper integrals​​.

Let's get our hands dirty. Consider a function $f(x)$ on an interval $[a, b]$. To approximate its area, we can slice the interval into a collection of smaller subintervals. This collection of slices is called a ​​partition​​, $P$.

On each little slice, say from $x_{i-1}$ to $x_i$, our function $f(x)$ will wiggle around. It will have a lowest point (an ​​infimum​​, which we'll call $m_i$) and a highest point (a ​​supremum​​, $M_i$). Now, we can build two kinds of rectangles on this slice.

1. A "pessimistic" rectangle with height $m_i$. Its area, $m_i(x_i - x_{i-1})$, is definitely less than or equal to the true area under the curve on that slice.
2. An "optimistic" rectangle with height $M_i$. Its area, $M_i(x_i - x_{i-1})$, is definitely greater than or equal to the true area.

If we sum up the areas of all the pessimistic rectangles across our partition, we get the ​​lower Darboux sum​​, $L(f, P)$. This sum is a guaranteed under-estimate of the total area. If we sum up all the optimistic rectangles, we get the ​​upper Darboux sum​​, $U(f, P)$, which is a guaranteed over-estimate. No matter what the "true area" is, it must be trapped between these two values:

Now, what happens if we make our slices finer? Imagine adding a new cut-point to our partition. The pessimistic estimate, $L(f, P)$, can only go up (or stay the same), and the optimistic estimate, $U(f, P)$, can only go down (or stay the same). They are being squeezed towards each other!

This leads us to a grand idea. Let's consider all possible partitions. The set of all lower sums has an upper bound (any upper sum will do!), so it must have a least upper bound. We call this the ​​lower Darboux integral​​, $\underline{\int_a^b} f(x) dx$. Symmetrically, the set of all upper sums must have a greatest lower bound. We call this the ​​upper Darboux integral​​, $\overline{\int_a^b} f(x) dx$.

These two values represent the best possible squeeze we can achieve. The lower integral is the highest we can push our under-estimate, and the upper integral is the lowest we can pull our over-estimate. The "true area" is still trapped between them:

So, when can we declare victory and say we have found the area? The answer is as simple as it is profound: we have found the area if, and only if, the squeeze is perfect. A function is ​​Riemann integrable​​ if its lower and upper integrals coincide. When they are equal, their common value is what we call the definite integral, $\int_a^b f(x) dx$.

This gives us a powerful test, the ​​Riemann criterion​​: a bounded function $f$ is Riemann integrable on $[a,b]$ if and only if for any tiny positive number $\epsilon$ you can dream of, it's possible to find a partition $P$ fine enough such that the gap between the upper and lower sums is smaller than $\epsilon$.

This is not just an abstract condition. It directly means we can make our approximation as accurate as we desire. If this gap can be made to approach zero, the lower and upper integrals must converge to the same value, and our function has a well-defined integral.

Let's take this principle for a spin and visit a "zoo" of functions to see which ones are tame enough to be integrated and which ones are too wild.

The Well-Behaved: Taming Jumps and Bumps

What if our function isn't perfectly smooth? Imagine we take a nice, continuous function like $f(x)=x^2$ on $[0,3]$, but we arbitrarily change its value at a single point. Let's say we declare that at $x=\sqrt{2}$, the value isn't $2$, but $0$. Have we broken the integral?

Let's think about the squeeze. When we calculate the upper sum, the supremum $M_i$ on any subinterval is unaffected by lowering a single point inside it. So the upper integral remains $\int_0^3 x^2 dx = 9$. What about the lower sum? For any partition, there will be at most one small subinterval containing the troublesome point $\sqrt{2}$. In that one interval, the infimum $m_i$ might drop to $0$. But this only affects one term in the entire sum! By making our partition finer and finer, we can make that one anomalous subinterval so narrow that its contribution to the gap between the upper and lower sums becomes vanishingly small. In the end, the lower integral also comes out to be $9$. The single "blip" was inconsequential.

This principle is quite general. You can have a finite number of "jump" discontinuities, like in a sawtooth wave, and the function remains perfectly integrable. The total area of the "bad" rectangles surrounding the jumps can be squeezed down to zero.

The Intractably Messy: The Dirichlet Catastrophe

So, what does it take for a function to not be integrable? We need the gap between the upper and lower sums to be stubbornly non-zero, no matter how fine our partition gets.

Consider the infamous ​​Dirichlet function​​, which takes one value for rational numbers and another for irrationals. Let's use a variation: on $[\pi, 2\pi]$, let $f(x) = 13$ if $x$ is rational and $f(x) = -5$ if $x$ is irrational. Both rational and irrational numbers are ​​dense​​, meaning you'll find them in any subinterval, no matter how tiny.

So, for any slice $[x_{i-1}, x_i]$ in any partition:

• The supremum, $M_i$, is always $13$.
• The infimum, $m_i$, is always $-5$.

When we calculate the upper sum, we are tiling the area with rectangles of height $13$, giving an upper integral of $13\pi$. When we calculate the lower sum, we are tiling with rectangles of height $-5$, giving a lower integral of $-5\pi$. The gap is permanent! The lower and upper integrals are miles apart ($18\pi$, to be precise), and the function is not Riemann integrable. The function is simply too "jagged" on too fine a scale. Similar issues arise for other functions that behave differently on rationals and irrationals, like one that is $x$ for rationals and $0$ for irrationals. The gap between the upper integral of $\frac{1}{2}$ and the lower integral of $0$ can never be closed.

The Surprisingly Tame: Miracles of the Popcorn Function and the Cantor Set

The story gets even more interesting. It's not just the number of discontinuities that matters, but their "nature."

Meet ​​Thomae's function​​, sometimes called the popcorn function. It's defined on $[0,1]$ as $f(x) = 1/q$ if $x=p/q$ is a rational number in lowest terms, and $f(x)=0$ if $x$ is irrational. This function is discontinuous at every rational point, yet it is Riemann integrable!

How can this be? The lower integral is easy. Since every interval contains irrational numbers, the infimum on any slice is $0$, making the lower integral $0$. The magic happens with the upper integral. The "pops" in the function are mostly very small. For any given height, say $\epsilon = 0.01$, there are only a finite number of rational points where the function value is greater than $\epsilon$ (only those with denominators less than 100). We can isolate all these "big" spikes within a finite collection of tiny intervals whose total width is as small as we please. On the rest of the interval, the function is very close to zero. By carefully choosing our partition, we can show that the upper sum can be squeezed arbitrarily close to $0$. Since both the upper and lower integrals are $0$, the integral of this bizarre function is $0$.

An even more profound example is the indicator function of the ​​Cantor set​​. The Cantor set is constructed by repeatedly removing the open middle third of intervals, starting with $[0,1]$. What's left is a "dust" of points which is uncountable, yet has a total "length" or ​​measure​​ of zero. If we define a function to be $1$ on the Cantor set and $0$ elsewhere, it seems like a mess. But again, the lower integral is clearly $0$. And because the total length of the intervals we remove from $[0,1]$ to create the Cantor set approaches $1$, we can construct partitions where the upper sum is built on intervals whose total length approaches $0$. The upper integral is also forced to be $0$. The function is integrable, and its integral is $0$.

The entire framework of Riemann integration is built on the idea of trapping a function in finite rectangles. This presupposes one crucial thing: the function must be ​​bounded​​. If we try to integrate a function like $f(x)=1/x$ on $[0,1]$ (setting $f(0)=0$), we immediately run into a problem. In the very first slice of any partition, $[0, x_1]$, the function shoots off to infinity. The supremum is infinite, making the upper sum infinite for any partition. The whole game collapses. This leads to the concept of improper integrals, a topic for another day.

Finally, the gap between the upper and lower integrals for functions like the Dirichlet type hints at a more powerful way to think about integration. Consider a function on $[0,2]$ that is $x^2$ for rationals and $2x-1$ for irrationals. Since $x^2 \ge 2x-1$, the upper Riemann integral will be $\int_0^2 x^2 dx = \frac{8}{3}$, and the lower integral will be $\int_0^2 (2x-1) dx = 2$. They don't match.

But what if we change the game? Riemann's method slices up the domain (the $x$-axis). Henri Lebesgue had a different idea: what if we slice up the range (the $y$-axis)? Instead of asking "for this vertical slice of $x$, what are the heights?", Lebesgue's approach asks "for this horizontal band of heights, what is the 'size' of the set of $x$'s that produce them?". The set of rational numbers has a "size" or measure of zero. The Lebesgue integral, in its wisdom, says that what the function does on this negligible set doesn't matter. It sees only the values on the irrationals. Therefore, the ​​Lebesgue integral​​ is simply $\int_0^2 (2x-1) dx = 2$.

The struggle to make the upper and lower Riemann integrals meet forces us to reckon with the structure of the number line itself—the density of rationals, the "size" of sets, and the nature of continuity. It sets the stage for a more general and powerful theory, a beautiful example of how grappling with the limitations of one idea can pave the way to a grander landscape of mathematical thought.

In our previous discussion, we uncovered the beautiful and simple idea at the heart of the Riemann integral. We imagined trapping the true, elusive area under a curve between two approximations: a lower bound built from rectangles that fit entirely underneath, and an upper bound made of rectangles that completely cover it. We declared victory—that the function is "integrable"—the moment these two bounds meet, squeezing the area into a single, unambiguous number.

You might be tempted to think this is a quaint mathematical exercise, a game of "what if" with little bearing on the real world. After all, the functions we meet in introductory physics or engineering—parabolas, sine waves, exponentials—are wonderfully well-behaved. For them, the upper and lower integrals snap together without any fuss. So, what is the point of this machinery? When does the gap between the upper and lower integrals actually matter?

This chapter is the answer to that question. We are about to embark on a journey where this very gap becomes our most insightful guide. It is a diagnostic tool that reveals the ragged edges of mathematical functions, showing us precisely where our methods work and where they break down. We will see that grappling with this "failure" is not a dead end, but a gateway to a deeper understanding of nature. It forces us to invent more powerful tools and reveals stunning, unexpected connections between calculus, probability theory, the physics of higher dimensions, and even the bizarre, infinite complexity of fractals.

Let's begin by meeting the ultimate troublemaker, a function so pathological it seems designed to break our integrators. Imagine a function that constantly flickers between two values, say $5$ and $-3$. The rule for its flickering is simple: if the input $x$ is a rational number, the function's value is $5$; if $x$ is irrational, its value is $-3$.

Now, try to apply our upper and lower integral strategy. Pick any small interval on the number line, no matter how tiny. Because both rational and irrational numbers are infinitely dense, packed together like two kinds of fine, intermingled dust, every interval will contain points of both types. So, for any rectangle in our partition, the lowest value (the infimum) is always $-3$, and the highest value (the supremum) is always $5$.

What happens to our sums? The lower sum, built from the infimums, will stubbornly calculate the area as if the function were always at $-3$. Over an interval of length $L$, the lower integral is $-3L$. The upper sum, built from the supremums, will calculate the area as if the function were always at $5$, giving an upper integral of $5L$. The gap between them is enormous and refuses to close, no matter how fine we make our partition. The function is not Riemann integrable. The integral, as Riemann conceived it, simply does not exist.

This reveals a profound principle: Riemann integration fails when a function oscillates too wildly, on a set of points that is "too big" and "too spread out." But what if we encounter something even stranger? Consider a special kind of fractal set, known as a "fat Cantor set." It's constructed by starting with an interval and repeatedly removing middle portions, but in a way that the total length removed is less than the whole interval. The result is a bizarre object: it's a cloud of disconnected points, containing no intervals at all (it is "nowhere dense"), yet it still possesses a tangible "size," or what mathematicians call a positive Lebesgue measure, let's say $\mu$.

Now, let's define a function that is $1$ on this fractal dust cloud and $0$ everywhere else. What are its upper and lower integrals? Since the set is nowhere dense, any partition interval will contain gaps, so the infimum of our function is always $0$. The lower integral must therefore be $0$. But the upper integral is a different story. To cover all the points of the set, our upper-sum rectangles must span the set's entire "size." It turns out that the smallest possible value for the upper sum is precisely the measure of the set, $\mu$, and thus the upper integral is $\mu$.

This a breathtaking result! The gap between the upper and lower integrals, $\mu - 0 = \mu$, is exactly the measure of the pathological set where the function "lives." The degree of non-integrability is not just some vague failure; it is a precise, quantitative measure of the size of the set causing the trouble. The gap tells a story.

Having stared into the abyss of non-integrable functions, let's now turn to cases where integration succeeds, often in the most surprising circumstances. This reveals a deeper, more nuanced story about what it takes for a function to be "well-behaved."

What if a function's discontinuities are not densely spread out? Consider a function that is mostly constant but has a few "jumps" at a finite number of points. In the physical world, this could represent a switch being flipped or a sudden force being applied. Here, the Riemann integral works perfectly. The reason is intuitive: we can isolate each of the finite jumps inside an interval that is as narrow as we like. The total length of these "quarantine zones" can be made vanishingly small. Inside these zones, the upper and lower sums may differ, but their contribution to the total integral becomes negligible as the zones shrink. Everywhere else, the function is smooth, and the upper and lower sums match perfectly. In the end, the total upper and lower integrals are forced to agree.

This is reassuring, but the truly remarkable cases arise when there are an infinite number of discontinuities. One might guess this is a death sentence for integrability, but that's not always so. Let's look at a function built by adding up an infinite number of small steps, one at each rational number. The function is discontinuous at every single rational point. And yet, if we construct it carefully so that it is always non-decreasing (monotonic), it turns out to be perfectly Riemann integrable! The gap between its upper and lower integrals is zero. Why? A monotonic function, even a jumpy one, is "tame." It is not allowed to oscillate wildly. By always heading in one direction, its jumps are constrained, and the integral machinery can handle it.

The most mind-bending example of all, however, may be the indicator function of the standard middle-thirds Cantor set. This function is $1$ on the Cantor set and $0$ elsewhere. The Cantor set is a monster: its points are totally disconnected, it has no interior, and it contains an uncountable infinity of points. Our function is discontinuous at every single one of them. Surely this function cannot be integrable. Yet, it is! Both its upper and lower integrals are zero.

The secret lies in the concept of "measure." Although the Cantor set has infinitely many points, it is "thin" like dust; its total length, or measure, is zero. This means we can cover the entire set with a collection of partition intervals whose total length is arbitrarily small. For the upper sum, this means the troubling regions can be squeezed into oblivion, forcing the upper integral down to zero. The lower integral is already zero, so they meet.

This example is the key that unlocks one of the deepest results in analysis: the ​​Lebesgue Criterion for Riemann Integrability​​. It states that a bounded function is Riemann integrable if and only if its set of discontinuities has measure zero. It doesn't matter if there are ten discontinuities or an uncountably infinite number of them. The only question is: what is their collective "size"? If it's zero, the Riemann integral exists.

The limitations of the Riemann integral, so clearly diagnosed by the gap between its upper and lower bounds, were not a failure for mathematics. They were an inspiration. They pointed the way toward a more powerful and general theory of integration.

Let us return to a cousin of our first troublemaker: a function that is $1$ for all irrational numbers and $0$ for all rational numbers. As we saw, its lower Riemann integral is $0$, and its upper Riemann integral is $1$. Riemann's method gives up, unable to provide a single answer. But our intuition screams for one. There are, in a very real sense, "more" irrational numbers than rational ones. The rationals form a countable [set of measure zero](@article_id:137370). So, our function is equal to $1$ "almost everywhere." Shouldn't its integral be $1$?

Enter Henri Lebesgue. He proposed a brilliant new way of thinking about integration. Instead of partitioning the domain (the $x$-axis) like Riemann, Lebesgue partitioned the range (the $y$-axis). His method, in essence, asks a different question:

• "For what values of $x$ is the function close to some value $y_1$?" Let's call this set $E_1$.
• "And for what values of $x$ is it close to $y_2$?" Call this set $E_2$. The Lebesgue integral is then roughly the sum of $y_1 \times (\text{size of } E_1) + y_2 \times (\text{size of } E_2) + \dots$.

For our function, the answer is simple. The function is equal to $1$ on the set of irrationals, which has measure 1. It is equal to $0$ on the set of rationals, which has measure 0. The Lebesgue integral is therefore $(1 \times 1) + (0 \times 0) = 1$. It gives a natural, robust answer precisely where Riemann integration failed. The study of when the upper and lower integrals differ directly motivated this more powerful theory, which has become the standard in modern physics, probability theory, and analysis.

The idea of generalizing the integral doesn't stop there. In the Riemann-Stieltjes integral, we introduce a second function, $\alpha(x)$, which acts as a "weighting function" or "integrator." Instead of summing up $f(x_i) \Delta x_i$, we sum $f(x_i) \Delta \alpha_i$, where $\Delta \alpha_i = \alpha(x_i) - \alpha(x_{i-1})$. This allows us to calculate weighted averages or integrals over non-uniform distributions. It finds profound use in probability, where if $\alpha(x)$ is the cumulative distribution function (CDF) of a random variable, the integral computes the expected value. But even this more general tool can fail, and once again, the test for its existence is whether its own upper and lower sums agree.

The world is not one-dimensional. The principles we have uncovered extend beautifully into the two- and three-dimensional spaces of our experience. Imagine a function defined over a unit square in the plane, which has a positive value $C$ only on the main diagonal line ($x=y$) and is zero everywhere else. What is its integral—representing, perhaps, the total mass of the square if its density is non-zero only along a thin wire?

The Darboux integral in 2D works with tiny rectangular volumes instead of intervals. For any such rectangle, its infimum is $0$ (since it must contain points off the diagonal), so the lower integral is $0$. What about the upper integral? The diagonal line, though containing infinitely many points, has zero area. We can cover it with a collection of thin rectangles whose total area can be made as small as we please. This forces the upper integral to be $0$ as well. They meet, and the integral is zero. This confirms our physical intuition: a line has no area, so an object whose mass is confined to a line has zero total mass in a 2D sense. This principle—that sets of lower dimension have zero measure in higher dimensions—is fundamental in physics when dealing with surface charges, line currents, and other boundary phenomena.

Finally, what does this theory say about the process of science itself? We often model complex systems by a sequence of simpler, solvable approximations. We hope that as our approximations get better, they converge to the true answer. The theory of uniform convergence gives us a wonderful guarantee. A key theorem states that if a sequence of Riemann integrable functions converges uniformly to a limit function, then that limit function is also guaranteed to be Riemann integrable. This is a profound statement about the stability of integration. It means that the property of being "well-behaved" is not fragile. We can have confidence that if our sequence of well-defined models is approaching a final state, that final state will also be well-defined.

The gap between upper and lower integrals, which began as a mere technicality, has led us on a grand tour of modern mathematics. It has served as a crucible, testing the limits of our functions and tools. It has shown us what lies beyond Riemann's framework and has illuminated the deep and beautiful consistency that underlies the mathematical description of our world.