Let $E$ be a vector bundle over a topological space $M$. I want to show that $\pi:E \to M$ is a homotopy equivalence.
To show this, I use the zero section $\zeta:M \to E$. Then $\pi \circ \zeta = Id_M$ and $\zeta \circ \pi$ is the zero map in the vector space $E_p$ for $p\in M$.
So the natural homotopy map is $H:E \times I \to E$ defined by $(v,t) \to t\cdot v$, where this operation makes sense if $v \in E_p$.
But I don't know how to confirm that this map will be continuous. It is continuous on each $E_x \times I \to E_x$ as it is just a scalar multiplication in the vector space, but how do I know that scalar multiplication is continuous on the entire total space of the bundle?