I'm having some difficulty comprehending the following Lemma:

let

be continuous at a point

in

. If

then there exist an

and a neighborhood U of c such that

.

now the proof that is given is:

let

. since

os continuous at

, there there is an

with a corresponding

such that

, then

, therefore

. If

, then

. If

then

, hence

. where

I don't comprehend what they are implying in the proposition, those it have something to do with the distance of f(x) going away from 0?

For the part in blue, why are they picking there epsilon as such?

For the part in red those it simply have to do with the fact that we are taking absolute values? I would think so, but am not 100% on that...