I am self studying vector calculus from Hubbard's book [I have no advanced background in topology, differential geometry and related subjects]. Here is the definition of a smooth manifold as per the text.
A subset M of R^n is a smooth k dimensional manifold if it is the graph of a C1 [continuously differentiable] function f expressing n-k variables as functions of the other k variables.
Here are my questions about this definition.
Why force the function f to be C1? Specifically, why do we need the derivative of f to be continuous? What happens if we allow differentiable functions whose derivative is not continuous?
As a follow up, a one dimensional smooth manifold is also called a smooth curve. For a (parameterised) curve to be smooth, is it really true that we require its tangent vector to be non-zero at all points? Do we lose the smoothness property even if the tangent vector at (at least) a single point is zero? Why or why not?
Thanks a lot for reading my question. I understand that I have asked a lot, but these things are bothering me.
Regards, Madhav