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I just want to know if I'm on the right track with these questions...

1. To prove that a point and a line in R^3 are not surfaces, I showed that the function from an open interval U in R^2 to the intersection of the point/line and a subset W in R^3 cannot be a homeomorphism. This is because the function is not injective.

2. For the curve y= sin(1/x), x does not equal 0, I want to know if the generalized cylinder on this graph is a smooth, regular surface.

Parametrized curve: (u, sin(1/u), 0)

So, sigma(u,v)=(u, sin(1/u), 0) + va

where a is a unit vector in the direction of the translation.

So, I showed that sigma is smooth on R/{0} by showing that all partial derivatives exist at all points except when u=0.

But, I'm having trouble showing the 'regular' part.

sigma_u = (1, -1/u^2 (cos(1/u)), 0)

sigma_v=a

So, for the surface to be regular, the cross product of the above two functions cannot be zero. So, the surface is regular if the tangent vector of the parametrized curve is not parallel to a.

I'm having trouble with this problem because by how the question is phrased, it seems like I'm only supposed to find a yes/no answer, not a 'depending on the vector a' answer. Do you see what I mean? So, I don't know if I'm missing anything, not proving rigorously enough... etc.

Any help would be greatly appreciated!