Haha another cool function – the Popcorn Function :
https://en.wikipedia.org/wiki/Thomae%27s_function
(Well-known for being continuous at any irrational number and discontinuous at any rational number)
Being continuous is a property that says that a function is in some sense "nice" near a point.
Being differentiable is also a "niceness" property at a point. In fact it is a much stronger property, in the sense that differentiability at a point implies continuity at a point.
It is clearly possible to find functions that are continuous but not everywhere differentiable (absolute value function fails to be differentiable at 0).
What is surprising (well, I find it surprising, and the discovery went against the beliefs at the time) is that there exist a function that is continuous everywhere and differentiable nowhere! : https://en.wikipedia.org/wiki/Weierstrass_function
In fact, in a certain sense, MOST continuous functions are differentiable nowhere!
Haha another cool function – the Popcorn Function :
https://en.wikipedia.org/wiki/Thomae%27s_function
(Well-known for being continuous at any irrational number and discontinuous at any rational number)
The concepts presented by uncountable nouns break down when applied to infinite sets. One must tread carefully and avoid fallacious conclusions when dealing with infinity. Remember to discern between uncountable and countable infinities.
Speaking of which, there is a one-to-one correspondence between the integers and the rationals. There are more transcendental numbers than algebraic numbers. And the set of all real numbers is exactly as big as the set of complex numbers.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
I don't think he was reminding you of those facts (countability of the rationals etc.); rather, I think he was adding them in as more 'interesting statements' for this thread. Or maybe you were referring to his first paragraph haha. Not sure in that case why (if at all that was his purpose) he was reminding you, maybe again just general statements for this thread.
I more meant that countability is kind of an irrelevant notion to the thing I mentioned so it seemed weird to bring it up as a reply to my post.
(And yeah, meant first para. Second para is good for this thread . Countability is such a nice and low-tech example of higher maths to show HS students.)
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
Proof using base arithmetic that Christmas is a pagan festival:
OCT 31 = DEC 25
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
Last edited by InteGrand; 1 Jan 2016 at 9:19 PM.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
I have always found fixed point theorems quite pretty.
Probably the most well-known one is the Brouwer fixed point theorem. One version of this states that if you have a continuous function f from a closed n-ball (eg the set of points of distance =< 1 from the origin in n-dimensional Euclidean space) to itself, then this function must have a fixed point, which is an x such that f(x)=x.
So in one dimension, this says that a continuous function f defined on the interval [0,1] that takes values in [0,1] must have a fixed point. (The 1-d version can be proved at high school level, try it!)
Fixed point theorems can have some pretty whack consequences. Eg, if I am in Russia and I put a map of Russia on a table, there will be a point on this map that lies directly above the actual physical spot in Russia that it represents.
They are also quite useful in abstract mathematics, for things like showing non-constructively that a certain equation/system of equations has a solution.
Last edited by glittergal96; 1 Jan 2016 at 11:51 PM.
Was the Brouwer fixed point theorem used by Nash to prove the existence of a Nash Equilibrium in a game or something? I don't know too much about game theory haha but this was something I think I heard. And is your use of Russia as the country at all a subtle reference to this game theory of Nash's time? Lol (seems too coincidental that you chose Russia )
Haha yes, spot on with it being a crucial ingredient in Nash's work. This is one of the applications in abstract maths I referred to in my last line which I added after you took your quote.
And nope, not an intentional reference. Russia was actually an arbitrary choice lol.
Last edited by omegadot; 30 Jan 2016 at 8:31 PM. Reason: Typo
Last edited by InteGrand; 5 Jan 2016 at 5:01 PM. Reason: Fixed LaTeX
100 people are standing on the positive real axis looking in the positive direction, each wearing hats coloured either black or white.
Each person can see infinitely far and hear from infinite distances.
Going in increasing order, these people are asked the colour of the hat on their head. (They are allowed to discuss a strategy before this whole guessing process starts).
It is slightly surprising that there is a strategy that guarantees correct guesses from 99 of these people.
What is slight more surprising is that these is still such a strategy if more hat colours are allowed (even an uncountable infinitude of hat colours!)
What is most surprising of all is that if there are countably infinite people in this line, and each of these people is deaf, AND there is an uncountable infinity of possible hat colours, we can STILL guarantee the correctness of all but finitely many guesses!!
This is another example of a consequence of the axiom of choice that some people often find unsettling. It also relies on the impossible assumption that a person can have infinite memory...ie they can recall infinite sets, functions on infinite sets, etc etc.
Removing the human / real world element though and viewing the result purely abstractly, it is still immensely weird.
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