 Solution

Take the Möbius line bundle over $S^1$,A continuous section of a vector bundle corresponds to a continuous real-valued function $s:[0,1]\to \mathbb{R}$ with $s(1)=-s(0)$. If $s(0)=0$, it already vanishes. Otherwise its endpoint values have opposite signs, so the intermediate value theorem gives $s(t_0)=0$ for some $t_0\in (0,1)$. Thus this vector bundle has no nowhere-zero section. Every continuous section must vanish.