Alright so I was doing homework for one of my math classes when I came across the problem
Let f be integrable on [a,b], and suppose g is a function on [a,b] such that g(x) = f(x) except for finitely many x in [a,b]. Show g is integrable and integral(f) = integral(g).
Now if you’re looking at this problem and you have no idea what it means or what it’s saying then you would not be alone because I looked at this and thought, “BROOOOOO come on now what in the hell does this even mean.” We have two functions here, they are the same function except for “finitely many points” where they are different, yet despite being different they also have the same integral?
Nonsense.
Absolute malarkey.
Let’s just think for a moment about what these two functions could possibly look like. This was my first thought
We take a coordinate plane, slap two functions on there that are together for a while, diverge momentarily, and then come back together. However, this is WRONG because of course there are infinitely many points in the interval where the two are separated because you can just keep taking smaller and smaller divisions. Alright then, if that’s the case then I guess we have to define our functions like this.
Where we have two functions that meet for their whole length except at finitely many (in this example one [1]) point discontinuities where the function g disappears for a moment and then comes right back. So we can see the structure of this right now, but I’m looking at it, like I’m looking at g and that does not look integrable to me, and even if it did, there is no way that their integrals would be exactly the same because that just wouldn’t make any sense. However, obviously, I’m the dumbass here, and whatever I think looks like makes sense is clearly wrong because that’s what the problem I’m solving says.
So, how do we prove something that I don’t even really believe is true? I bang my head against the metaphorical math wall until I start seeing things that aren’t there and this problem starts to make sense.
DARBOUX INTEGRALS [2]
SO we have these things that exist, they’re called Upper and Lower Darboux Summations, they look like this
The notation makes them look MUCH more complicated than they actually are, but that doesn’t matter. What’s important for you to know is that when the Upper Darboux Sum equals the Lower Darboux Sum, the value that they share is equal to the integral of the function.
Cool?
Cool.
So, with these things established as our tools for integration, we can now define a new function h(x) = f(x) – g(x). This means that h looks like this
Where it equals zero at every point except x0 where it takes its one and only non-zero value. Having a (almost except at that one point) zero function is good because it standardizes our notation for any possible function meaning our result is always true. It is also helpful because the Lower Darboux Sum of this function equals 0 [3]. The Upper Darboux Sum is a little trickier to prove, but the basic argument is that the amount of width that that discontinuity has is infinitely small because it is just one singular point, so the area underneath it is also infinitely small and the Upper Darboux Sum agreed with the Lower Darboux Sum so that they share a value of 0.
This is good because it means that we’re basically done. The Upper and Lower Sums agree so we know that
integral(h(x)) = 0
And by the definition of h, we know that
integral(h(x)) = integral(f(x) – g(x)) = integral(f(x)) – integral(g(x)) = 0
Which finally tells us that
integral(f(x)) = integral(g(x))
Which is the result we were trying to prove and ba-da-bing ba-da-boom, we’re done. AND IT ONLY TOOK ME 800 WORDS
[1] – The induction argument for why this can be represented as just one discontinuity instead of like a bajillion is really annoying so I’m just not gonna talk about it.
[2] – The stuff I wanna write about is annoying and I don’t have the word count to really explain everything I’m talking about, so I have three options. Either I stop talking about math, I keep apologizing for not properly explaining stuff, or I just decide that explaining things is for professors and I’m just here to complain. I’ve decided to try and stay in between options two and three for a compromise between your sake and mine.
[3] – Trust me bro