Does a continuous function take Cauchy sequences to Cauchy sequences?

Let $(X,d)$ be a metric space. A function $f:X\rightarrow X$ maps Cauchy sequences to Cauchy sequences. Then if $x_n\rightarrow x$, then $\{x_n\}$ is Cauchy, implying $\{f(x_n)\}$ is Cauchy. But if $X$ is not complete then $\{f(x_n)\}$ may not converge, let alone converging to $f(x)$. So the condition doesn't imply $f$ is continuous.

However what about the converse? If $f$ is continuous does it take Cauchy sequences to Cauchy sequences?

No. Take $\iota\colon(0,+\infty)\longrightarrow\mathbb R$ defined by $\iota(x)=\frac1x$ and the sequence $\left(\frac1n\right)_{n\in\mathbb N}$.