Can I use variable substitution on $\delta$-$\epsilon$ proofs?

Question

When I try to prove the continuity of a function $f:\mathbb{R}\to\mathbb{R}$ at $a$, can I make the substitution $x = a+y$, and thus:

$0 < |x-a| < \delta \implies |f(x)-f(a)|<\epsilon$

becomes:

$0 < |y| < \delta \implies |f(a+y)-f(a)|<\epsilon$

I noticed that this simplifies things for some functions.

Answer 1

It's just a name of a variable. Name it whatever you like. If you subsitute $y=x-a$ again, you get just $0<|x-a|<\delta \implies |f(a+x-a)-f(a)|=|f(x)-f(a)|<\epsilon $ again. I don't see any problem here. Do you have an example where you noticed something weird after substituting or where the proof significantly changed?

Answer 2

Yes, you may .

The two statements $$0 < |x-a| < \delta \implies |f(x)-f(a)|<\epsilon$$

$$ 0 < |y| < \delta \implies |f(a+y)-f(a)|<\epsilon$$ are equivalent.