Sunday, July 3, 2016

real analysis - Equivalent definitions of Continuity in BbbR

I have some questions on logical implications of the definition of continuity. Here is the context:




  1. Let (R,d) a metric space with the standard metric.

  2. Define f:RR to be a function, and let
    x0R

  3. f is said to be continuous at x0 [ϵ>0,δ>0;xR,|xx0|<δ|f(x)f(x0)|<ϵ].




My question is: Are the following implications true? I feel like I am mixing up some stuff.



1) f is continuous at x0 [ϵ>0,δ>0;xR,|xx0|<δ|f(δ)|ϵ]. (So that it is sufficient to construct a δ satsifying |f(δ)|ϵ).



2) f is continuous at x0 |f(xx0)|f(x)f(x0) (I have a feeling I am mis-using the topological definition of continuity.



PARTIAL END: NO NEED TO READ BEYOND (but if you are feeling inspired, then please continue :)






If 1) and 2) are true, then to prove continuity, can we do something along the lines of the following:? (Please forgive the complete lack of rigor, I am just trying to run things on a probably false sense of intuition)




  • Fix ϵ=ϵ0>0


  • We wish to find a δ>0 such that xR,|xx0|<δ|f(x)f(x0)|<ϵ0.


  • But, there exists g(x)R:g(x)|xx0|=|f(x)f(x0)|


  • But $g(x)*|x-x_0|=|f(x)-f(x_0)|



  • And g(x)|xx0|=|f(x)f(x0)|<ϵ0


  • So we wish to find a δ>0 such that δϵg(x)




If my last part gives no insight however wrong it may be, please tell me.

No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...