Why can't all pointwise continuous functions preserve Cauchy sequences?

Question

If $f$ is a real-valued uniformly continuous function on $A$, then for every Cauchy sequence $(x_n)$ in $A$, $(f(x_n))$ is a real Cauchy sequence.

But why do we need the uniform continuity of $f$ for this? If $f$ is continuous on $A$ then for any point $c$ in $A$, and any Cauchy sequence $(x_n)$ converging to $c$, $(f(x_n))$ converges to $f(c)$, thus making it a Cauchy sequence.

A continuous function maps convergent sequences to Cauchy sequences. But - for general domain $A$ - not all Cauchy sequences are convergent. — Daniel Fischer, Jan 27 '18 at 15:52

score 2 · Answer 1 · answered Jan 27 '18 at 15:52

2

Take $f(x)=\dfrac{1}{x}$ on $(0,\infty)$ which is continuous but not uniformly and $a_n=\dfrac{1}{n}$

answered Jan 27 '18 at 15:52

Mostafa Ayaz

33,056

Oh. So if the point of convergence of $(a_n)$ lies outside the domain we may face this issue. – Smooth Alpert Frame Jan 27 '18 at 15:55
In fact we need compactness in this case and what you argued is true... – Mostafa Ayaz Jan 27 '18 at 15:57

Why can't all pointwise continuous functions preserve Cauchy sequences?

1 Answers1