Defining and Understanding Continuous Unit Normal Fields on Orientable Surfaces

[JG89]

So I've been reading about orientated surfaces lately, and I always see the definition that a surface S is orientable if it is possible to choose a unit normal vector n, at every point of the surface so that n varies continuously over S.

However, what does "varies continuously" mean? I never see this statement made precise and it is ambiguous (to me at least)

 

[Hurkyl]

It means if you write n as a function of P (the point on the surface to which n is normal), that function is continuous.

 

[JG89]

So let's take the Mobius Strip. If p is any point on the strip and n is a unit normal to p, and we transverse the strip and come back to the same point p, then we end up with the unit normal -n, where we should have had it as n, since the unit normal should have been continuous, right? And this is our contradiction?