I have some questions on logical implications of the definition of continuity. Here is the context:
- Let (R,d) a metric space with the standard metric.
- Define f:R→R to be a function, and let
x0∈R - f is said to be continuous at x0 ⟺[∀ϵ>0,∃δ>0;∀x∈R,|x−x0|<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;∀x∈R,|x−x0|<δ⟺|f(δ)|≤ϵ]. (So that it is sufficient to construct a δ satsifying |f(δ)|≤ϵ).
2) f is continuous at x0 ⟺|f(x−x0)|≤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 ∀x∈R,|x−x0|<δ⟹|f(x)−f(x0)|<ϵ0.
But, there exists g(x)∈R:g(x)∗|x−x0|=|f(x)−f(x0)|
But $g(x)*|x-x_0|=|f(x)-f(x_0)|
And g(x)∗|x−x0|=|f(x)−f(x0)|<ϵ0
So we wish to find a δ>0 such that δ≤ϵg(x)
If my last part gives no insight however wrong it may be, please tell me.
No comments:
Post a Comment