Suppose that f is continuous on (a,b), if f is also differentiable on (a,b), expect possibly at a single point x0 where lim(x--x0)(f'(x)) exists and is finite. prove that f is diff at x0 and that f' is cont at x0.
can anyone solve this? i have no idea when i processed to lim(f'(x)).. thanks!
Some (including me) would prefer using directly the mean value theorem. It is really straightforward.
Let us denote (assumed to exist). Let . Choose such that and imply . Then, for all such , the mean value theorem provides such that , hence because . This proves that is differentiable at and . The assumption then reads " is continuous at ".