Let $f(x)=\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x}$ if $0<x \leqslant \pi$ and $f(x)=0$ if $x=0$.
I took the sequences $$x_n=\frac{1}{n}$$ $$y_n=\frac{1}{n^2}$$
We have that $x_n-y_n \rightarrow 0$ but $$f(x_n)-f(y_n)=\frac{1}{2}( \sin{ \frac{1}{2n}}-\sin{ \frac{1}{2n^2}}) +(n^2-n)$$ does not converge to $0$.
Is this solution correct?
Thank you in advance!