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.
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.