In section 2.2 on "What is a manifold?" under
equation (2.8) he says ф(x)=|x

I had to think about that one and had a quick look at this video on the Khan academy to refresh my memory. I then got to use my own graph paper to test the assertion for myself. What fun! Now I have put it in Excel. Here is the result.

^{3}| is a C^{2}function because it is infinitely differentiable everywhere except at x=0 where it is differentiable twice but not three times.I had to think about that one and had a quick look at this video on the Khan academy to refresh my memory. I then got to use my own graph paper to test the assertion for myself. What fun! Now I have put it in Excel. Here is the result.

ф(x)=|x

^{3}| is in blue. It's gradient is obviously negative for x<0 and positive for x>0.
So

For x<0, ф'=-3x

^{2}and for x>=0 ф'=3x^{2}. The gradient of ф' is always positive.
So ф''=|6x|. We can now see the problem at x=0 in green.

It is hardly necessary to plot ф''' which is -6 for x<0,
6 for x>0, but undefined at x=0.

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