Vector field that vanishes in a non empty open set of $S^2$.

Question

I have to find a non-trivial vector field $V$ on the emty sphere $S^2$ which vaishes in a non-empty open set of $S^2$. I tried something as $V(x,y,z)=0$ if $z<0$ and $V(x,y,z)=(\text{something that has length }z)$ otherwise. In this way $V$ is continuos but it is not of class $C^\infty$. What am I doing wrong?