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FYI, (A) is not ''smooth'' at certain points while (B) is ''smooth''
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Thanks @Jemurray3 ! I discovered that I asked a silly question!

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Is that direction right? or wrong?

The problem is that.. how to find the limit at that point? :(
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Is discontinuous, as is its derivative.

at x=0? or generally speaking?

just at the origin

Yep.

It depends on what function you were modeling in the bottom left. What was it?

I'm sorry but I don't know. It's just an example provided by mathmate which I'm having trouble with.

@rolypoly @jemurray3
Here are some further information for the lower-left example (cusp).
An example would be
\( \large f(x)=10-x^\frac{2}{3} \)
Continuity, which is necessary for differentiability, is satisfied here, since f(0) \( \in R \) and is finite, and f'(0+)=f'(0-).
However f(x) is not differentiable because Darboux's theorem requires that, for f(x) to be differentiable between a and b, f'(x) must take on every value between f'(a) and f'(b). If we set a=\( -\epsilon \) and b=\( +\epsilon \), where \( \epsilon \) is an arbitrarily small number \( \ne 0\), we see immediately that Darboux's theorem is not satisfied, and therefore f(x) is not differentiable at x=0.
This is a case that illustrates that continuity is a necessary but not sufficient condition for differentiability.

The definition of a limit implies a finite quantity.