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A continuous function $f : \mathbb{R} → \mathbb{R}$ is uniformly continuous if it maps Cauchy sequences into Cauchy sequences.

is the above statement is true?

I guess it is not true but can't find any counterexample.

user58266
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  • No, $f(x)=x^2$ is a counterexample, and I think there might be a duplicate question. However, I apologize for my hasty vote to close, because I voted to close as duplicate of the following question which is not a duplicate (without reading carefully): http://math.stackexchange.com/q/40676/ – Jonas Meyer Jan 14 '13 at 16:59
  • (Pete L. Clark's answer at the linked question points out why every continuous function $\mathbb R\to\mathbb R$ maps Cauchy sequences to Cauchy sequences.) – Jonas Meyer Jan 14 '13 at 17:05

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The answer is no as explained by Jonas Meyer and every continuous function $f:\mathbb R\longrightarrow \mathbb R$ has this property:

If $(x_n)_{n\in\mathbb N}$ is a Cauchy sequence then $m\leq x_n \leq M, \ \ \forall \ n\in\mathbb N$ for some $m<M$. Since $f$ is uniformly continuous on $[m,M]$ the result follows.

So $f(x)=x^2$ (or any continuous not uniformly continuous function $\mathbb R\to \mathbb R$) is a counterexample.

P..
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    Alternatively, because every Cauchy sequence in $\mathbb R$ converges, there exists $x$ such that $x_n\to x$, and by continuity $f(x_n)\to f(x)$. Since $(f(x_n))_n$ converges, it is a Cauchy sequence. – Jonas Meyer Jan 19 '13 at 06:21
  • @JonasMeyer: Right, that is better. – P.. Jan 19 '13 at 06:25
  • @JonasMeyer you were talking about the proof of converse in the comment, right? –  Nov 01 '17 at 01:25
  • @Maneesh: No, the comment is about an alternative proof that every continuous $f:\mathbb R\to \mathbb R$ sends Cauchy sequences to Cauchy sequences. – Jonas Meyer Nov 01 '17 at 01:46
  • sorry! actually, I didn't see the continuity of $f$. –  Nov 01 '17 at 03:57
  • Question was this no?A continuous function $f:\mathbb R→\mathbb R$ is uniformly continuous if it maps Cauchy sequences into Cauchy sequences. –  Nov 01 '17 at 03:58
  • @JonasMeyer your comment was cauchy sequence under continuous map is a cauchy sequence.Right? –  Nov 01 '17 at 04:00
  • sorry! I misunderstood the question. –  Nov 01 '17 at 04:03
  • Now I understood. Thank you very much :). –  Nov 01 '17 at 04:04