# Continuous images of Cauchy sequences are not necessarily Cauchy

Could you please provide an example for two metric spaces $X,Y$, a continuous function $f$ that maps $X$ to $Y$ and a Cauchy sequence in $X$, which is not mapped to a Cauchy sequence in $Y$ by $f$?

Does $f(x) = \frac{1}{x}$ work if $X$ is any metric space and $Y$ is the set of real numbers?

• No, it doesn't work for any space, but you just need one example right? Take $X = (0,1), Y = \mathbb{R}$ and $f : x \mapsto 1/x$ – Prahlad Vaidyanathan Oct 22 '13 at 17:03
• A function between metric spaces is called Cauchy-continuous if it takes Cauchy sequences to Cauchy sequences. Every Cauchy continuous function is continuous. The converse holds if $X$ is complete. – Stefan Hamcke Oct 22 '13 at 17:54
• If $f$ is uniformly continuous between metric spaces, then Cauchy sequences are mapped to Cauchy sequences. – Ron Apr 30 '17 at 15:52

$X=(0,1),Y=\mathbb{R},f(x)={1\over x}, {1\over n}$ is cauchy in $X$ but $f({1\over n})=n$ which is not cauchy in $\mathbb{R}$