1. Basic:
|y|= 0 or > 0 for all y
2. Limit: $latex displaystylelim_{xto a}f(x) = L$ ; x≠a
|x-a|≠0 and always >0
hence
$latex displaystylelim_{xto a}f(x) = L$
$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
$latex boxed{|x-a|<delta}$
$latex implies |f(x)-f(a)|< epsilon$
Mathtuition88 