Uniformly continuous functions and the preservation of Cauchy sequences

Question

A function which maps a metric space into a metric space is continuous if and only if all convergent sequences are mapped to convergent sequences. Can we think a uniformly continuous function as the one which preserves “cauchy-ness” of a sequence, that is every Cauchy sequence is mapped to Cauchy sequence? Please elaborate more on this.

Answer 1

No, the two concepts are not equivalent. For example, the function $f(x) = x^2$ on the real line preserves Cauchy sequences, but is not uniformly continuous. Indeed, any continuous function from $\mathbb{R}$ to $\mathbb{R}$ preserves Cauchy sequences because in $\mathbb{R}$, being a Cauchy sequence is equivalent to being convergent, and continuous functions preserve convergence.

References:

• How do I prove a uniformly continuous function preserves Cauchy sequences?
• If continuity preserves convergence, and Cauchy sequences are convergent sequences, why do we need uniform continuity to preserve Cauchy sequences?