Prove or disprove by counterexample that if $\sum_{k=1}^\infty x_k$ exists and $\lim_{k\to\infty}f(x_k)=0$, then $\sum_{k=1}^\infty f(x_k)$ exists

Question

For all positive integer $k$, let $x_k\in\mathbb{R}$. Assume that $\sum_{k=1}^\infty x_k$ exists. Let $f:\mathbb{R}\to\mathbb{R}$ and assume $\lim_{k\to\infty}f(x_k)=0$. Can we prove or disprove by counterexample that $\sum_{k=1}^\infty f(x_k)$ exists?

I suspect there are obvious counterexamples for this but I am not sure how to find any. Also, I am interested to know if there are conditions on $f$ (e.g., continuous, differentiable, bounded,...) such that $\sum_{k=1}^\infty f(x_k)$ exists. Any help is appreciated.

Answer 1

Counterexample.

Let $x_k=\frac1{k^2}$ and let $f$ be prescribed by $x\mapsto\sqrt x$ on $[0,\infty)$ and $0$ elsewhere.

Answer 2

Other counterexample: $x_k=\frac{(-1)^k}{k}$ and $f(x)=|x|$