εδ Confusion in Limit & Continuity

1. Basic:
|y|= 0 or > 0 for all y

2. Limit: $latex \displaystyle\lim_{x\to a}f(x) = L$ ; x≠a
|x-a|≠0 and always >0
hence
$latex \displaystyle\lim_{x\to a}f(x) = L$
$latex \iff $
For all ε >0, there exists δ >0 such that
$latex \boxed{0<|x-a|<\delta}$
$latex \implies |f(x)-L|< \epsilon$

3. Continuity: f(x) continuous at x=a
Case x=a: |x-a|=0
=> |f(a)-f(a)|= 0 <ε (automatically)
So by default we can remove (x=a) case.

Also from 1) it is understood: |x-a|>0
Hence suffice to write only:
$latex |x-a|<\delta$

f(x) is continuous at point x = a
$latex \iff $
For all ε >0, there exists δ >0 such that
$latex \boxed{|x-a|<\delta}$
$latex \implies |f(x)-f(a)|< \epsilon$