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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.

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  • $\begingroup$ Such condition is weaker than uniform continuity. It is called Cauchy continuity. See en.wikipedia.org/wiki/Cauchy-continuous_function $\endgroup$ – Crostul Apr 20 '18 at 17:14
  • $\begingroup$ Wait so it’s a necessary condition for a mapping to be uniformly continuous. $\endgroup$ – John Mitchell Apr 20 '18 at 17:16
  • $\begingroup$ Yep: it is necessary, but not sufficient. In general , for a map between metric spaces you have $$\mbox{uniform continuous $\Rightarrow$ Cauchy continuous $\Rightarrow$ continuous}$$ $\endgroup$ – Crostul Apr 20 '18 at 17:17
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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:

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