If $x_i(t)$ $i=1,..,n$ ($n=2,3$) regular functions($k$ times continuously differentiable functions) and $\sum_{i=1}^n(x_i^\prime(t))>0$ $n=2,3,t\in(a,b)$ then
$x_i=(x_i(t))$ is equation for regular cuve.
Now proof starts with saying that we need only to prove that it is local bijective map.But we need also that inverse of map to be continuous as well.I don't know why author does not prove this or maybe this follows from given conditions which I can't understand.