This is the proposition and proof saying that if we know that S is a regular surface and x is a candidate of parametrization which satisfies 1)differentiable 2)surjective differential map 3)continuous bijective then, x has a continuous inverse thus resulting that x is a real parametrization
However, in the proof, I cannot see that where I used the condition that S is a regular surface
Please let me know where the condition S is a regular surface used and is there any counter-example that x does not have a continuous inverse when S is not a regular surface?
condition 1 in the proposition is that x is differentiable
condition 2 is that locally homeomorphism
condition 3 is that differential dx is one-to-one