= Solution
Take the \b[<Möbius line bundle> over $S^1$],
$$
L=([0,1]\times\mathbb R)/((1,t)\sim(0,-t)).
$$
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 \b[this vector bundle has no nowhere-zero section]. Every continuous section must vanish.
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