Who decided the statements: >"For a function to be integrable,

They dont.

They don't mean the same thing actually. The first statement says that it is necessary for the function to be continuous for it to be integrable. That means if you take some random function and want to know if you could integrate it you need to check if it is continuous. If it is not, you can't integrate it since that it needs to be for it to be integrable. But the first statement doesn't say that continuity is sufficient for integrability, maybe the function needs to satisfy some other condition besides continuity. According to the first statement it may be possible to find some continuous function that isn't integrable, it doesn't specify if continuity on its own is enough to tell if a function is integrable.

The second statement on the other hand says that if the function is continuous it will always be integrable which is the same as what the third statement says.

>>9099550
Yeah, I fucked up that example, but in general:
>if P then Q
>Q, provided that P
>For P, it is necessary that Q
all mean the same thing, and it's really confusing.

>>9099535

I can give you the perspective from logic:
If (a function is continuous) then (it is integrable).
(Function is integrable) only if (a function is continuous).
Now these two statements are logically equivalent.
However the first statement you offered doesnt seem to be logically equivalent with the others because you can infer from other two statements that a function being continuous is (sufficient) condition for a function to be integrable.
While the first statement says that a function being continuous is necessary condition for a function to be integrable.

English is easy to learn and hard to master

>>9099582
This is what brainlets actually believe.

>>9099557
that's not true either, none of those are identical statements

>>9099598
Wheres your proof for such an outrageous claim?

>>9099605
>if P then Q
does not say that P is the only way for Q to exist, it just means that P is one way to get to Q. P can happen at any time, and if P happens, Q is also happening

>For P, it is necessary that Q
says that Q must be the case before P happens. Q can happen at any time, and if it does, one of the conditions for P has been met.

>>9099598
They are. Welcome to the real world where people are stupid enough to use "if" in definitions instead of "iff."

>>9099620
even if we discount the first statement for use of loose ifs, the last two are still different. They're going different directions.

>>9099619
"If P then Q" is same as either "P is sufficient for Q" or "Q is necessary for P" or both.

https://en.wikipedia.org/wiki/Necessity_and_sufficiency

>>9099557
>>9099535
Are you trying to pretend that your language doesn't have multiple ways of saying the same thing?

>>9099619
found the brainlet. it's you.
P => Q means precisely that Q is a necessary condition for P

>>9099535
>"For a function to be integrable, it is necessary that it is continuous"
not true, there are discontinuous integrable functions

>>9099535
They're all saying the same thing, just in different ways. The concept is that a function needs to be continuous for you to be able to integrate it.

Statement 1 says that if you want to be able to integrate a function, it must be continuous.

Statement 2 says that you can integrate a function if it is continuous.

Statement 3 says that if a function is continuous then you can integrate it.

>>9099878
Only when you can break the function into discrete continuous sections. You can't integrate, for example, a function that is 1 on every rational point, and 0 on every irrational point.

>>9099535
like the rest of the thread said, they don't mean the same thing

1) if you want X, then you need Y. however, perhaps you need more than just Y for X to happen, but you absolutely know X cannot happen without Y: Not Y -> Not X
2) X absolutely happens if Y: Y -> X
3) If Y then X: Y ->X

suppose you want to bake a cake:
1)For you to be able to bake a cake, it is necessary that you have milk
you CANNOT say "i have milk, therefore i can bake a cake" based off that statement alone; if you have milk you might be missing other shit like eggs and flour.
you CAN say "i don't have milk, therefore i can't bake a cake"

the law of contrapositives lets you do this with that statement though:
We know No Milk -> No Cake
Lets say we made a cake. Obviously that means that every ingredient needed for the cake was available, including (but not limited to) milk
Logically this means Cake -> Milk
Not Y -> Not X is the same as X -> Y

the statements you have actually prove bidirectionality
1 is X->Y
2 and 3 are Y -> X
they compound together to create X if and only if Y, or X <--> Y

>>9099619
>can happen at any time
>before
That's not how logic works, nigger.