The same pathetic proof that mixes apples and oranges, and makes the reader believe that MVT has anything to do woth FTC!
Then came some back-and-forth. I argued that there are many approaches, and due to different motivations we’ve chosen different ones. Michael argued that I was part of some “Church of Limitology” which “indoctrinates” calculus students, and that my proofs “suck”, are “trashy”, and are “in bad taste”. My point that this is not the approach I actually take in a classroom setting was ignored.
However, he did have some points. One of them was that we can weaken the theorem to only assume that the function is continuous at the point , and my proof assumed way too much to say the function is continuous everywhere. But let’s consider where I actually used continuity. First it shows up in invoking the Integral Mean Value Theorem, but there I really only need it to say there’s a maximum and a minimum, so the Darboux sums work out. An integrable function still manages to satisfy this condition. Then I use continuity to show that , which really only needs continuity at . In fact, my proof already works in Michael’s extended context.
He also tried presenting his own proof of the crucial step. He argues by continuity at that for any there is a so that when . This fact then shows that , and so the limit (there’s that awful word again!) that I claimed in the post works out.
But what does his proof really mean? Go on, try and draw a picture. It’s saying that the difference in area we add by integrating from to from the area we’d add if we just used a constant height of by less than any constant multiple of the width . And that means… it’s obscure to me, at least.
On the other hand, my proof says that the area function changes by some amount as we go from to , which means there’s some average (“mean”) rate of change over that interval. At some point along the way, the derivative actually attains that mean value, and as we contract the interval we push that middle point down to . Now that makes sense to me.
Now, the real genius came later, after I sewed the two parts of the FToC together. Eventually, Michael said:
By MVT I meant the one for continuous function, that it hits the zero if it changes sign. Is it the one you were talking about, or you were talking abouth the Lagrange theorem (= MVT for the derivative)? I’m a bit confused. Well, either way it’s not too important.
And here it all runs off the rails. This whole time he hasn’t actually been reading a single thing about my proof, and evidently he hasn’t read the proofs in the calculus textbooks he so despises. He doesn’t even know which theorem I’m invoking! And it is important, because the different theorems say vastly different things.
Now it’s plain as day that Michael is a crank, pushing his pet theories while remaining so embittered to the “system” that “indoctrinates” students against him. Either that or he’s been trolling. The actual merits of my own proof — which I hope I’ve shown above to meet his tests — never mattered at all. I do hope he will set up his own weblog to present all his work in his own space, and then interested readers can judge for themselves the merits and demerits of different proofs.
As for this discussion, it’s closed. I have a proof, and Michael has a proof, and they both work. Our proofs emphasize different aspects of the theorem, and we choose between them depending on what we want to highlight for our current audience. Despite all his ranting, neither one is “the right way” or “the wrong way”, independent of context. I’m glad to hear alternative approaches here, since they might highlight points that I missed. But as a word to future ranters: don’t even try to use my weblog as your soapbox. That sort of behavior really is trashy, and in bad taste.
Besides, I’m the Dennis Miller around here.
[UPDATE]: I’ve come to a decision, since the war seems to rage on unabated, and Mr. Livshits refuses to take the olive branches of equanimity I’ve been offering since the beginning. As of midnight (Central Standard Time) tonight, this is over. Mr. Livshits goes in the kill file, and I wash my hands of the whole business. I’m sure he’ll cry foul, and oppression, and maybe he’s right. However, this whole mess just distracts from my work here, and I’m sick of it. From sideline conversations with numerous non-commenting readers, I’m not the only one.
I’ve made my case, and tried over and over to say that ultimately the whole debate comes down to aesthetics. His approach has its merits, as does mine. He really dislikes my approach, so much so that he’s willing to fight tooth and nail. Ultimately, I really don’t care to fight this any more. But since I can’t seem to continue this project without having my “mathematical taste” insulted left and right, I’m using my authority as the owner of this space to cut off debate. This does not continue here.
If Mr. Livshits wants to continue his tirades, he’s free to set up his own weblog, as I’ve encouraged him to do time and again. He can even continue to read and post to his own space in parallel to my coverage. If he’s right and a significant majority of my readers want to hear his side, he’ll have a built-in audience ready and waiting, and he’s welcome to it. Just like the sky, there’s a lot of blogosphere out there. Of course, he’ll eventually have to come up with something to fill his space, since I’m not spending the rest of my life here on calculus and elementary analysis.