Does $\lim |a_n-b_n|=0$ imply $\lim |f(a_n)-f(b_n)|=0$?

Question

Let $f:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ a continuous function and $(a_n)$, $(b_n)$ two sequences with values in $\mathbb{R}_{\geq 0}$ and $\lim |a_n-b_n|=0$. Does this imply $$\lim |f(a_n)-f(b_n)|=0?$$ Hint: the sequences $a,b$ needn't to be convergent.

Althoug I know this statements holds for uniformly continuous functions, I think it is not true for continuous functions generally. But no counterexample comes into my mind. Has anyone an idea?

Answer 1

Consider $a_n=\ln(n+1)$, $b_n=\ln(n)$. Then $|a_n-b_n|=\ln(1+1/n)\to 0$, but with $f(x)=e^x$ we find $|f(a_n)-f(b_n)|=1 \to 1$

Answer 2

Take $f:x\mapsto x^2$, and for $n\ge 1$ have $a_n=n+\frac{1}{n}$ and $b_n=n$.

Then $|a_n-b_n|=\frac{1}{n}\to 0$ but $|f(a_n)-f(b_n)|=2+\frac{1}{n^2}\to 2$.