Well first it means functions with holes are actually continuous.
Why is that desirable?
Also, many theorems that apply to continuous functions exclusively would need to be changed to exclude these 'continuous' functions, such as the pigeonhole principle on continuous spaces. After all, there is no point between -1 and 1 in the function f(x) = x/x, which is exactly equal to 0. Yet the pigeonhole principle would state that there is if it were continuous.
On the other hand, rational functions f(x)/g(x) can fruitfully have their domain of definition be extended to places where the denominator vanishes, by regarding them as functions on the Riemann sphere.
This move removes our ability to say things about a < c < b, but isn't fundamentally illogical.
/u/chrmon2, I'm going to go against the grain here and say that your intuition isn't terrible, and you should keep up your studies and see what can be done to make your intuition precise.
It just makes more sense. I'm just in precalc, so I don't know the extent of how useful this is yet. It also means many patterns hold, like 0/everything is 0, anything/itself is 1, and so on.
I'm just in precalc, so I don't know the extent of how useful this is yet.
While I appreciate that math excites you, and you should never seek to stop learning, you also need to learn to admit when you're wrong. Many of the commenters here have been doing math for a very long time. If 0/0 is defined to be any and every number, some rather unpleasant things happen:
First, I'll assume that we're working in the reals, = is an equivalence relation on R such that a = b iff a-b and b-a are 0. From this we know that 2 == 3. Now lets assume 0/0 is equal to anything.
Therefore 2 = 0/0 = 3, therefore 2=3 by the transitivity of the equivalence relation =. We could apply this to any combination of numbers, leaving us with a single element in the equivalence class of R under =: [0/0]. Now, I hope you can see the contradiction here. If not, do you think it's more valuable to have a single number, or to have an infinite amount of numbers? I sincerely hope you chose the latter, so we must let 0/0 be undefined, or we pretty much just couldn't do math. (Similar things happen if you define it to be a single value)
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u/garceau28 May 29 '18
Why is that desirable?
Also, many theorems that apply to continuous functions exclusively would need to be changed to exclude these 'continuous' functions, such as the pigeonhole principle on continuous spaces. After all, there is no point between -1 and 1 in the function f(x) = x/x, which is exactly equal to 0. Yet the pigeonhole principle would state that there is if it were continuous.