What does it mean for a function to be Riemann integrable?

Question

I need to come up with a precise mathematical definition of what a Riemann integrable function is. I know what the Riemann integral is but when I look for definitions all I find are proofs of how to prove that a function is Riemann integrable. I need help creating a definition of what it means for a function to be Riemann integrable that does not include any notation, just a couple of mathematical sentences that defines Riemann integrals.

Answer 1

The Riemann integral of a function on $[a,b]$ is the limit of Riemann sums whose partitions $[a,b]$ get finer and finer (i.e. the norm of the partition goes to zero). If this limit exists, then the function is said to be Riemann integrable and the value of the Riemann integral is the limit the sums approach.

Answer 2

The Riemann integral is defined in terms of Riemann sums. Consider this image from the Wikipedia page:

We approximate the area under the function as a sum of rectangles. We can see that in this case, the approximation gets better and better as the width of the rectangles gets smaller. In fact, the sum of the areas of the rectangles converges to a number, this number is defined to be the Riemann integral of the function.

Note however that we can draw these rectangles in a number of ways, as shown below (from this webpage)

If, no matter how we draw the rectangles, the sum of their area converges to some number $F$ as the width of the rectangles approaches zero, we say that the function is Riemann integrable and define $F$ as the Riemann integral of the function. For some functions the area will not converge, the canonical example being the indicator function for the rationals $\mathbb{1}_{\mathbb{Q}}(x)$, which is $1$ if $x$ is a rational and $0$ otherwise.

Answer 3

A positive function is Riemann integrable over the interval $[a,b]$ if the infimum of the upper sums equals the supremum of the lower sums. (You'll have to look up what an upper sum and lower sum are. An upper sum intuitively is an approximation of the area of the curve from above, while a lower sum is an approximation of the area of the curve from below.)

In other words, the best approximation of the area of the function from below equals the best approximation of the area of the function from above.

Answer 4

Maybe an historical perspective will help. It never hurts to have many points of view on what a mathematical definition is doing.

For most of the 18th century the integral was considered just an antiderivative, much the way that many calculus students still consider it. If you want to compute $$ \int_a^b f(x)\,dx $$ you really must find an antiderivative $F$ for the function $f$ and then write or compute $$ \int_a^b f(x)\,dx =F(b)-F(a).$$ Cauchy in the early 19th century felt that this needed to be placed on a more rigorous footing.

If you assume that the function $f$ is continuous and that continuous functions have antiderivatives then take any points $a=x_0<x_1<x_2<x_3< \dots < x_n =b$ and use the mean-value theorem to select points $\hat x_i\in (x_{x-1},x_i)$ for which $$f(\hat x_i)\cdot(x_i-x_{i-1})=F(x_i)-F(x_{i-1}).$$ If you carefully check the very simple arithmetic you will get that $$ \int_a^b f(x)\,dx =F(b)-F(a) = \sum_{i=1}^n f(\hat x_i)\cdot(x_i-x_{i-1}).$$ Cauchy's bright idea was to notice that, while it would be hard to figure out the exact points $\hat x_i$ that make this work you can use, instead any other point $\xi_i\in [x_{i-1},x_i]$ as long as $f(\hat x_i)$ and $f(\xi_i)$ are close together. For continuous functions this is easy to arrange. You don't get an exact formula for the integral, you get an approximate formula: $$ \int_a^b f(x)\,dx =F(b)-F(a) \approx \sum_{i=1}^n f(\xi_i)\cdot(x_i-x_{i-1}).$$

The end result is that Cauchy proved that the integral of every continuous function could be approximated by these Riemann sums.

Riemann just asked the obvious question:

Are there other functions (not just continuous) that also have an integral using Cauchy's same method.

So the class of Riemann integrable functions is the class of functions for which Cauchy's method works. This is somewhat larger than the class of continuous functions, large enough a class that nineteenth century mathematicians thought that they had a pretty good theory of integration. (They didn't.)

Answer 5

To answer this question, I will assume you already know what the integral of a step function is. First I will answer with notation, and then give an equivalent formulation without notation.

Let $f \colon [a,b] \to \mathbb{R}$ be a function. We say that $f$ is Riemann-integrable if for every $\epsilon > 0$, there are step functions $\varphi$ and $\psi$ on $[a,b]$ such that

1. $\varphi \leq f \leq \psi$ and

2. $\int (\psi - \varphi) \leq \epsilon$.

Without notation: A function on a closed bounded interval is called Riemann-integrable if there are two step functions respectively above and below it, and these two step functions can always be chosen so as to make the area between their graphs less than any specified quantity.

Answer 6

Here is the way that I first encountered it (somewhat nonstandard): Let $f:[a,b] \rightarrow \mathbb{R}$ be a function, and $[a,b]$ be a non-degenerate closed bounded interval. Let $P$ is a partition of $[a,b]$ if $P=\{x_{0},x_{1},...,x_{n}\}$ is a set such that $a=x_{0}<x_{1}<...<x_{n}=b$. Then let the mesh of $P$ be defined by $||P||=max\{x_{1}-x_{0},x_{2}-x_{1}...,x_{n}-x_{n-1}\}$.

A representative set of $P$ is $t=\{t_{1},...t_{n}\}$ such that $t_{i} \in [x_{i},x_{i-1}]$ for all $i \in \{1,...,n\}$.

This representative set is necessary because riemann integrality is going to ultimately depend (intuitively) on a bunch of little rectangles. In computation, we pick either the left side of each rectangle or the right side in a Riemann sum. But, actually to prove integrability, we need to go further and say that the function's integral converges to a value, independently of how we choose these rectangles. Further, the mesh is important, because the rectangles need not be of equal width, all we need is that all of them are sufficiently "skinny." This means that the widest rectangle is still really "skinny." How skinny?

Let $P$ be a partition of $[a,b]$ and $t$ be a representative set. $f$ is Riemann-integrable if there exists some $K \in \mathbb{R}$ and $\delta>0$ such that $||p||<\delta$ imply that $| \sum_{i=1}^{n} f(t_{i})(x_{i}-x_{i-1})-K|<\epsilon$ for each $\epsilon>0$.

Please note that $K=\int_{a}^{b}f(x)dx$.

This is the maximum amount of generality (that I know of) to show that something is Riemann integrable without invoking upper and lower sums. It does not depend on our choice of "rectangles", since the representative set is arbitrary, and it only requires that the "widest" rectangle is smaller than some real number.

The difficulty with this definition is that you need to have a guess for what the integral actually is before proving that the function is integrable. This is why upper and lower sums are actually so important, and why most definitions require them. It can be proven that Darboux integrability implies Riemann-integrability.

For further references or further study, you can refer to Ethan Bloch's "The Real Numbers and Real Analysis."

Without additional definitions, it is probably easier to invoke some vague notions of the Darboux integrable. Assuming some familiarity with the concept: there exists a partition of $[a,b]$ such that $|U(p,f)-L(p,f)|<\epsilon$ for any $\epsilon>0$. even more loosely stated: the riemann sums of $f$ on a non-degenerate closed-bounded interval converge to a value as you take the sum over a sufficiently large number of subdivisions of the interval.