No. The relationship between epsilon and delta can be very complex depending on f. Epsilon measures the variation of f(x) whereas delta measures the variation of x.

I don't see how this follows.

The difficulty here may be in using the same variable name

in two different statements. Let me rewrite this.

We are given

for every

there exists a

such that for all

,

implies

. (**)

We need to show

for every

there exists a

such that for all

,

implies

. (*)

We start the proof in the standard way. Fix an arbitrary

. Instantiate

to

in (**). Then (**) says that there exists a

such that

implies

. So, take

. Then for every x, if

, then

, as required to prove (*).