MHB Continuous and differentiability

Joe20
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Hello,

I have attached the question and the steps worked out. I am not sure if my steps are correctly. Need advise on that.
Next, I am not sure how to show f''(0) exist or not. Thanks in advance!
 

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You have shown that $f$ is differentiable at $0$ and that $f'(0) = 0$. For $x\ne0$ you can use the usual rules for differentiation, to see that $f'(x) = 2x$ if $x>0$ and $f'(x) = -2x$ if $x<0$. Therefore \[f'(x) = \begin{cases}2x&\text{when }x\geqslant0,\\-2x&\text{when }x<0.\end{cases}\] Now you have to decide whether that function is differentiable at $x=0$.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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