Equivalent definition of uniform continuity at infinity

Question

I am trying to make an equivalent definition of uniform cointiniity for functions that converge at infinity. Thank you in advance for your time.

Given $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x)\rightarrow l$ when $x\rightarrow \infty$

*The traditional definition of uniform coninuity that I know:

$f$ is uniformly continuous iff for every $\epsilon>0$ there is a $d>0$ such that for every x,y (pertaining to $\mathbb{R}$) $|x - y|< d$ implies $|f(x) - f(y)|< \epsilon.$

*This are two definitions that I would like to prove to be true (given same $f$) :

1) $f$ is uniformly continuous iff for every $\epsilon>0$ there is an $M>0$ such that for every $x,y$(pertaining to $\mathbb{R}$) $ x,y>M$ implies $|f(x) - f(y)| < \epsilon$

2) $f$ is uniformly continuous iff for every $\epsilon>0$ there is an $N>0$ such that for every two sequences $\{X_n\}$,$\{Y_n\}$ that diverge to $(\infty$) $Ñ>N$ implies $|f(X_ñ) - F(Y_ñ)|< \epsilon$

Are 1) or 2) equivalent to the traditional definition of uniform continuity?

Answer 1

Since you only consider asymptotic behavior of the function as $x\to +\infty$, I assume you actually mean to study functions $f: R^+\to R^+$, since for functions on $R$ itself one can't conclude anything without knowing something about the asymptotic behavior at $-\infty$.

If so, the functions $f$ you consider, namely those that satisfy $\lim_{x\to\infty}f(x)=1$, are necessarily uniformly continuous. Your conditions (1) also guarantees that there will be a limit value at infinity, and therefore the function is uniformly continuous.

Your condition (2) is a bit problematic because $N$ should be dependent on the sequences chosen (you must change the order of the quantifiers).

Answer 2

no, there could be an N such that after some well-described first upper limit, the functional behavior approaches abruptly a differing limit; the upper bound could be a spike jump (upwards or downwards) from the previous lower/upper limit due to accumulated built-up potential. Thus, the continuity constraint plays a key-role. Does it infer to monotonic or non-monotonic functions, that's the main question to consider. For ratioanl functions that could be an evasive question.