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I just started learning differential geometry. Got some questions. Thanks in advance to anyone who can help!

Consider the one-dimensional manifold represented by the line y = x for x<0 and y = 2x for x>= 0. Now if I consider the altas with two charts p(x, y)=x for x<-1 and q(x,y)=y for x>-2. The transition map is C-infinity in (-2,-1).

Next I consider another altas with r(x,y)=x for x<1 and s(x,y)=y for x>-1. The transition is not differentiable at 0 inside (-1,1).

Can I say I've defined two differential structures of the same manifold?

But what does it mean by a one-deminsional manifold having only one differential structure up to diffeomorphism?

Thanks!

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# Differentiable manifold

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