clarification of continuous functions and Cauchy sequences A continuous function need not preserve a Cauchy sequence. For instance,
$f : (0,1) \to (e,\infty)$ where $a_n = \frac{1}{n}$ and $f(x) = e^{1/x}$
If $f$ is uniformly continuous then it will preserve.
My question is:
If the domain is complete or the co-domain is complete does this have an effect on preserving a Cauchy sequence?

Continuous functions preserve Cauchy sequences if their domain is complete. If that's the case and $x_n$ is a Cauchy sequence in the domain, then $x_n$ converges to some $x$ in the domain. But then due to continuity, $f(x_n)$ converges to $f(x)$ and is thus a Cauchy sequence.
The codomain has no influence on this. For a counterexample where the domain is incomplete and the codomain is complete, yet a continuous function doesn't preserve Cauchy sequences, take any arbitrary continuous function which doesn't preserve Cauchy sequences, and replace its codomain by its own closure. The function still doesn't preserve Cauchy sequences, even though the codomain is now complete.
