Riemann Integration: Part 2
Let us continue our discussion about Riemann Integration!
Now at this point, we have a notion of what our upper and lower sums looks like and what they tell us about a function, but how do we know that these upper and lower sums even tell us anything accurate about our function? Well, we can utilize a very powerful result that comes from a long list of inequalities. Now, you can prove this but for the sake of time, I'm just going to show it. Given a function f and an arbitrary partition, P we have that
Essentially, what this tells us is that if our upper and lower sums approach each other, we can be sure the the Riemann Integral of our function exists. This is a very powerful result in that now we can prove that a function is Riemann Integrable. In other words a function is Riemann Integrable iff:
What a useful result. Now, given any function f, we can show it is Riemann Integrable so long as we can cleverly choose a Partition so that the above inequality is true.
In part 3, we will use this inequality to prove some incredibly funky functions are integrable. Stay tuned!