Ufyukyu

up vote 1 down vote favorite How would I give a simple example of a function $f$ differentiable in a deleted neighborhood of $x_0$ such that $lim_{xto x_0}f^prime(x)$ does not exist? I cannot seem to think of an example. A delete neighborhood is an open interval about $x_0$ which does not contain $x_0$ . So, $(x_0-delta,x_0+delta)-{x_0}$ for some $delta>0$ . How would something be differentiable in a deleted neighborhood if at the point of the derivative, the limit does not exist. Presumably, the derivative ends up looking something like $lim_{xto x_0} dfrac{1}{x}$ , if it does not exist. calculus derivatives examples-counterexamples share | cite | improve this question