Is a Function on Bounded Domain Taking Cauchy Sequences to Cauchy Sequences Uniformly Continuous?

I have a question similar to this one: Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?

So, from that discussion, we know that a function that takes Cauchy sequences into Cauchy sequences must be continuous, yet such function is not necessarily uniformly continuous. For example, we consider x^2 function on real numbers. The image of any Cauchy sequence is also Cauchy, where the function is not uniformly continuous.

So can we put some conditions on the domain of the function which takes Cauchy Sequences to Cauchy Sequences so that the function is uniformly continuous. We already have that it is continuous, so if the domain is a closed and bounded, we get uniformly continuity. Yet if the domain is open and bounded, do we get this result?

So, if this result does not hold, can you give a counter example or a proof if the result holds?

Any kind of hint or help is appreciated.

• In $\mathbb R$, you can use the following: If the domain is bounded then its closure is compact. Since your function maps Cauchy to Cauchy you can extend it continuously to its closure and apply already known facts. – Tim B. Dec 27 '16 at 16:57