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Let $(X,d), (Y,\delta)$ be metric spaces. Then we know the following:

Proposition $1$. A function $f:X\rightarrow Y$ is continuous (by definition, reflects open sets to open sets) iff maps convergent sequences to convergent sequences.
Proposition $2$. If a function $f:X\rightarrow Y$ is uniformly continuous (by $\epsilon-\delta$ definition), then it is continuous.
Proposition $3$. If a function $f:X\rightarrow Y$ is uniformly continuous, then it maps Cauchy sequences to Cauchy sequences. The converse is false: given that $\mathbb{R}$ is a complete metric space, there are continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ which are not uniformly continuous, such as $e^x$.

Indeed, if the converse of Prop. $3$ were true, then we would get an interesting result, by rephrasing Prop. $2$:

Proposition. $4$. If a function $f:X\rightarrow Y$ maps Cauchy sequences to Cauchy sequences, then it does the same with convergent ones.

Due to this, the converse of Prop. $4$ is false.

QUESTION. Now, if $Y$ is also complete, then Prop. $4$ is obviously true. How about in general? If it is true, then we have a new form of continuity (preserving Cauchy sequences), between continuity and uniform continuity. I suppose it is false, but I can't come up with a counterexample, since I don't know many metric spaces which are not complete. Thank you in advance!

Paul Rebenciuc
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    You received this from who? Is there some context? Is there some reason you want the solution, despite knowing nothing about the problem? – xxxxxxxxx Apr 18 '21 at 14:37
  • It certainly look like an ODE, but only the person who asked you knows for sure what he meant. (Unless it's a homework problem he doesn't understand.) If he won't clarify it, I would just forget about it. – saulspatz Apr 18 '21 at 14:38
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    As you seem to have the problem statement there typed out already, I am not understanding what you are asking. – xxxxxxxxx Apr 18 '21 at 14:40

1 Answers1

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Prop 4 is true. Suppose $f$ maps Cauchy sequences to Cauchy sequences, and let $x_n$ be a sequence that converges to $x$. Consider the sequence $x_1, x, x_2, x, x_3, x, \ldots$ consisting of $x_n$ with $x$ interspersed. This is also Cauchy, so the sequence $f(x_1), f(x), f(x_2), f(x), f(x_3), f(x),\ldots$ must be Cauchy. But that implies $f(x_n)$ converges to $f(x)$.

Robert Israel
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