Close graph of multivalued function

Question

Assume that $X,Y$ are two closed sets in $R^n$ (with the induced euclidian topology) and let $f:X \rightarrow Y$ be a multivalued function. Assume also that the graph of $f$ is a closed subset in $X \times Y$ is the following true?

If $\lim_{n\rightarrow \infty}x_n= x_0$ then $\lim_{n\rightarrow \infty}f_*(x_n)$ (where $f_*(x_n)\in f(x_n)$) exists and converges for some $y\in f(x_0)$?

Thanks

Answer 1

This is not true. Let $X=Y=[0,1]$ and let $$f(x)=\begin{cases} [0,1] \mbox{ for } x=0\\ \left\{ \sin x^{-1} \right\} \mbox{ for } 0<x\leq 1\end{cases}$$ then consider a sequence $x_{n}=\left(\frac{\pi}{2} + n\pi\right)^{-1} .$