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When a question states that "how smooth is $f~$" what should I do? For example how can we apply it in this question :

Consider the function $f: R\rightarrow R$ defined by

$ f(x) = \begin{cases}x^{2}\sin⁡(1/x^2 ) & ~~\text {if}~~ x≠0;\\ 0 & ~~\text{if}~~ x=0; \end{cases}$

How smooth is $f$ ? What should I do? Is green theorem related to these kind of questions?

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Smooth function – pedja Feb 17 '12 at 7:00
up vote 2 down vote accepted

$$f(x) = \begin{cases}x^{2}\sin⁡(1/x^2 ) & ~~\text {if}~~ x≠0;\\ 0 & ~~\text{if}~~ x=0; \end{cases}$$

$$f'(x) = \begin{cases} 2x\sin⁡(1/x^2 ) - \frac{\cos(1/x^2 )}{x} & ~~\text {if}~~ x≠0;\\ 0 & ~~\text{if}~~ x=0; \end{cases}$$

If you see the graph, Smoothness Graph, the function is clearly not continuous at x=0.

For more details on how to proceed, check this.

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It means that you should investigate the following:

Is $f$ continuous? Is $f$ differentiable? If $f$ is differentiable, then is the derivate $f'$ a continuous (or differentiable) function? If $f'$ is differentiable, then what about $f''$? Etc.

No Green's theorem that I know of is even remotely related to this.

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