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|
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.
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
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.
@@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.
@@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 .
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?
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 :)
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.
