There is actually another way of proving continuity of x^2, i.e. that the limit of x^2 = a^2 as x goes to a. |x^2 - a^2| = |x - a| * |x + a| = |x - a| * |x - a + 2a|
@oa_math
2 жыл бұрын
this is way better, thank you!
@goodplacetostop2973
4 жыл бұрын
12:38 12:00 “Because 0 is not equal to 1” Proof ? 😛
@j9dz2sf
4 жыл бұрын
If 0=1 then Ø={Ø}, then Ø has an element (Ø), which is a contradiction :-)
@tomkerruish2982
4 жыл бұрын
R is a field. By definition, the additive and multiplicative identities of a field are distinct. Somewhat less flippantly, the Peano axioms suffice, since 0 is not the successor of any natural number, while 1 is the successor of 0.
@DeJay7
4 ай бұрын
@@j9dz2sf "If 0=1 then Ø={Ø}" That's even less convincing than just "0 ≠ 1" Jokes aside, if you really wanna prove that 0≠1, then yes, that's probably the way to go, since I think one of the most fundamental definitions of numbers are in terms of elements in sets. Correct me if I'm wrong.
@j9dz2sf
4 ай бұрын
@@DeJay7 Yes, in Set Theory ZF, everything is a set: 0 is defined as Ø, 1 is defined as {Ø}, 2 is defined as {Ø,{Ø}} and so on en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers#Definition_as_von_Neumann_ordinals .
@douglasstrother6584
5 ай бұрын
My Calculus Professor (Tony Tromba, UC Santa Cruz 1981) dropped the Dirichlet Function on us at the end of a Friday lecture to give something to discuss at Happy Hour.
@SimplyChrisRLP
4 жыл бұрын
This was amazingly accessible, thank you. Could you do a video explaining constructive logic and how to prove there? How to rationalize sequential continuity without L.E.M
@sadececansu9
Жыл бұрын
Thanks a lot for this marvelous video!!! The second example was the one what I was looking for
@yoav613
4 жыл бұрын
in the first example you can take delta to be -a+sqrt(a^2+e) and than you have (x-a)(x+a) is less than a^2+e-a^2=e.
@sirlight-ljij
4 жыл бұрын
Here is a nice related problem: it can be shown that function f(x)=0, x is irrational, 1/n if x=m/n, gcd(m,n)=1 is continuous in all irrational points and not continuous in all rational. The question therefore is: is there a reverse function, not continuous in all irrational points and continuous in all rational?
@xutzl4925
4 жыл бұрын
I've found very interesting and brilliant equation that I can't solve. The command for the task: Solve cos(cos(cos(cos(x)))) = sin(sin(sin(sin(x)))). x is a real number. This problem comes from Russian Math Olympiad, 95. Michael, I believe you can overcome this task :)
@chaitanyagaur7928
4 жыл бұрын
here is an interesting fact cosx function iterated even times and sinx funtion iteterated for same number of times are not equal for any real x
@chaitanyagaur7928
4 жыл бұрын
@@angelmendez-rivera351 what can you say about general case?
@JorgeGomez-li9td
4 жыл бұрын
Very cool, thanks for the quality in all the aspects!!!
@SonVu-t1j
Жыл бұрын
Why you dont choose sqrt(-epsilon + a^2)
@xaxuser5033
4 жыл бұрын
when i saw the function that u gonna present in the video i was genuinely amazed
@JonathanMandrake
3 жыл бұрын
I already knew the function, but yeah, it´s quite clever!
@mathunt1130
3 жыл бұрын
One of the questions we had to answer was: Show that the following function is continuous at 0. f(x)=x for x in Q or f(x)=0 otherwise.
@ryanschade8001
Жыл бұрын
I am working on this problem now. At x=0, using his proof with our definition of f(x)=x, our f(xo)=f(a) at zero so it is continuous at that point only. I believe this works for the definition of continuity which uses neighborhoods of epsilon around f(xo) as well.
@arnoldvillodas4997
4 жыл бұрын
Excelente video , gracias por hacerlo.
@QuantumHistorian
2 жыл бұрын
Can someone point me to the "last video" referred to in the intro? I'm on the real analysis play list and I can't find it anywhere.
@brucedienst7553
2 жыл бұрын
i can't either :(
@ArthurDetaille
4 жыл бұрын
Baby Penn be like : *High pitched sound*
@НиколайШерстюк-ы7е
4 жыл бұрын
2:08 f(x)=x^2 is continous all all [sic!]
@kevinmartincossiolozano8245
4 жыл бұрын
Could you do a video about uniform continuity? Thanks :)
@MichaelPennMath
4 жыл бұрын
That is upcoming, maybe in another week I will be at that section.
@kushagrasinghal8209
11 ай бұрын
very helpful.
@kumaralok2098
3 жыл бұрын
very awesome can u make video on dirichlet and Thomas fun pls reply
@pandas896
4 жыл бұрын
I beg , please someone tell me , what is real analysis?
@commenting000
4 жыл бұрын
It's really just centered around mastering the art of proving things.
@malawigw
4 жыл бұрын
@VeryEvilPettingZoo some people still doesn't know about google ...
@elshaddai225
4 жыл бұрын
If 'n'th is an odd possitive integer, prove that coefficients of the middle terms in the expansion of (x+y)^n are equal......
@cerwe8861
4 жыл бұрын
Should be easy using pascals triangle (and that it's symmetric)
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