When is a function non-differentiable?

[damabo]

I know that $e^{rx}$ is an infinitely differentiable function.
However, say you have f= x. this is clearly one time differentiable, giving 1. a second time it can be derived as well, giving 0. is 0 differentiable, indeed differentiation gives zero.
So when is a function non-differentiable. I'm thinking of cusps, such as on x=0 with the function f=|x|, or other functions where the derivative is undefined. correct? are there other cases? it would also be interesting to see a function that is only finitely differentiable.

 

[pwsnafu]

The Weierstrass function is the most famous example.

 

[damabo]

that's a nice one!

 

[chiro]

Hey damabo.

There is a definition in analysis that is based on showing the various limits from both sides exist and its written in what you call the delta epilson model.

You have this:

http://en.wikipedia.org/wiki/Differentiable_function#Differentiability_in_higher_dimensions

and you also the definition for a limit to exist:

http://en.wikipedia.org/wiki/(ε,_δ)-definition_of_limit

But you need to consider this in terms of general norms and not simply absolute values.

 

[damabo]

thanks, will apply that formula.