Fibre bundle over product space

Question

Let $p:E\to A\times B$ be a continuous map, from which I want to check that it’s a fibre bundle with fibre $F$. If I knew that $E|_{A\times b}$ and $E|_{a\times B}$ are fibre bundles with fibre $F$ for every $a\in A$ resp. $b\in B$, would this be enough? Are there any counterexamples?

Answer 1

This is false: Take your favorite discontinuous function $f=f(x,y), f: R^2\to R$, which is continuous in each variable $x, y$. Let $E\subset R^3$ denote the graph of $f$ and let $p: E\to R^2=A\times B$ be the projection to the xy-plane. Then $p$ is not a fiber bundle but over each line $A\times \{b\}$, $\{a\}\times B$, the map $p$ is a fiber bundle.