Consider the function $$ h:V_{i}\times\mathbb{R} \rightarrow V_{i}\\ ([z],t) \mapsto [z_{0}:\ldots:tz_{i}:\ldots:z_{n}] $$ where $V_i=\mathbb{C}P^n\backslash\{0\}$. I want to prove that this function is continuous. I first thought that I could just write $h$ as a composition of the quotient map, a map in $V_i$ and the inverse of the quotient map. Something like this:
$$ h([z],t)=[z_{0}:\ldots:tz_{i}:\ldots:z_{n}]=\left(q_{n}\circ\alpha_{t}\circ q_{n}^{-1}\right)\left([z_{0}:\ldots:z_{i}:\ldots:z_{n}]\right) $$ But this seems fishy: I am not sure of the continuity of $q^{-1}$ and I have "hidden" the variable $t$ in $\alpha$ (and I do not know if I can do it).