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

$latex boxed{|x-a|<delta}$

$latex implies |f(x)-f(a)|< epsilon$