When is a function not differentiable?

Question

anonymous · Answer

when the limit of the difference quotient does not exist.  there could be many different reason why it would not exist. there is not one answer

anonymous · Answer

Thanks, but what does it mean when a function IS differentiable?

anonymous · Answer

the derivative is a limit 
it is 
$$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$$ if for some values of x this limit does not exist, then the function is not "differentiable" for that value of x

anonymous · Answer

and if that limit does exist, then it is differentiable for that x.  but there can be several reasons why the limit may not exist

anonymous · Answer

1) limit from the left is not the same as the limit from the right as in 
$$f(x)=|x|$$ at 
$$x=0$$

anonymous · Answer

2) the function grows too rapidly at a point for example 
$$f(x)=\sqrt[3]{x}$$ at 
$$x=0$$

anonymous · Answer

3) the function is not even defined at the point for example 
$$f(x)=\frac{1}{x-2}$$ at 
$$x=2$$

anonymous · Answer

but if the derivative DOES exist at a point, it means in some sense the function is "smooth" there. continuous for sure, but also no corners