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.
Real Analysis – What Does It Mean for a Function to Be Riemann Integrable?
real-analysisriemann-integration
Best Answer
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.