I want to prove the following property:
Let $V,W$ be quasi-projective varieties, let $W=\bigcup\limits_{i\in I}W_i$ be an open covering and let $f_i:f^{-1}(W_i)\rightarrow W_i$ be induced by $f:V\rightarrow W.$ Then $f$ is regular iff $f_i$ are regular for all $i\in I.$
What I did: let $\varepsilon_i:f^{-1}(W_i)\rightarrow V$ be an inclusion. Then $f_i=f\circ\varepsilon_i$ is regular as the composition of regular mappings.
How can we establish the inverted implication? I am stuck a bit...
I suspect what you mean by 'regular' is 'morphism' of quasi-projective varieties. (Hartshorne Chapter I, section 3, Definition on page 15)
To show $f$ is a morphism: First clearly $f$ is continuous.
Let $F\subset W$ open and $F\to k$ be a regular map (locally it looks like a fraction of polynomials of same degree etc.). We want to show $f^{-1}(F)\to F\to k$ is also regular.
Let $p\in f^{-1}(F)$. $p\in f^{-1}(W_i)$ for some $i$. Now $f_i$ being a morphism means it takes the regular function $F\cap W_i\to k$ to a regular map $\gamma_i: f^{-1}(F\cap W_i)\to F\cap W_i\to k$. So there is a neighbourhood $E\subset f^{-1}(F\cap W_i)$ containing $p$ such that $\gamma_i: E\to k$ is a given by a fraction of polynomials of same degree. Now $E$ is open neighbourhood of $p$ in $f^{-1}(F)$, such that $E\subset f^{-1}(F)\to F\to k$ regular.
So we are done.