Is this function differentiable at zero?

I'm having a bit of trouble deciding whether functions like this are differentiable so steps would be helpful.

Thanks.

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- April 20th 2010, 01:32 PMfeatherboxIs this function differentiable at zero?
Is this function differentiable at zero?

I'm having a bit of trouble deciding whether functions like this are differentiable so steps would be helpful.

Thanks. - April 20th 2010, 01:49 PMLaurent
Since the value at 0 is 0, this function is differentiable at 0 if and only if has a limit as . In the case of , the ratio equals if and if (because then ), hence

- it converges to 0 if ;

- if , it converges to 1 from the right and -1 from the left, hence it doesn't converge

- if , it diverges to from the right and from the left, hence it doesn't converge.

As a conclusion, it is differentiable iff , and the derivative is 0 then.