Extending a local smooth section to a global one

Question

Let $\pi:E\to B$ be a smooth vector bundle and suppose $U$ is an open set of $B$. Let $\sigma:U\to E$ be a local smooth section of $\pi$. Can we extend $\sigma$ to global smooth section $\tilde{\sigma}:B\to E$? Assume that $B$ is paracompact.

Answer 1

Consider the trivial bundle $p:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ defined by $p(x,y)=x$. Let $f(x)=(x,{1\over x})$ the section on $\mathbb{R}-\{0\}$, you cannot extend it to $\mathbb{R}$.