geometrically, the function #f# is differentiable at #a# if it has a non-vertical tangent at the corresponding point on the graph, that is, at #(a,f(a))#. That means that the limit

#lim_{x\to a} (f(x)-f(a))/(x-a)# exists (i.e, is a finite number, which is the slope of this tangent line). When this limit exist, it is called derivative of #f# at #a# and denoted #f'(a)# or #(df)/dx (a)#. So a point where the function is **not** differentiable is a point where this limit does **not** exist, that is, is either infinite (case of a vertical tangent), where the function is discontinuous, or where there are two different one-sided limits (a cusp, like for #f(x)=|x|# at 0). See definition of the derivative and derivative as a function.