It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff
$$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X \quad x\_{n} \rightarrow p \Rightarrow f(x\_{n}) \rightarrow f(p)$$


It is also well known that if $X$ and $Y$ are metric spaces and $f : X \rightarrow Y$ is uniformly continuous, then $f$ maps Cauchy sequences to Cauchy sequences. 


By analogy it seems plausible that if a function between metric spaces maps Cauchy sequences to Cauchy sequences then it must be uniformly continuous. However mimicking the proof of the analogous result for continuous maps doesn't work, which makes me think the result if false. Does anyone know any counterexamples?


Also on the uniform continuity wikipedia page, it says that the result is true if $X$ and $Y$ are subsets of $\mathbb{R}^{n}$. EDIT: It actually doesn't say this, I misread the page.



Does Cauchy continuity imply uniform continuity? [No.]
No it's not true.


f(x) = x^2 on whole real line.


It maps Cauchy sequences to Cauchy sequences but it's not uniformly continuous on the whole real line.


