There is still one more thing you need to show. Remember the theorem says that if exist and are continous and is continous then is differenciable. You want to show that is continous at any point . Say that then . And so . We want to show we can make this sufficiently small in the punctured disk . We can safely ignore the case when because then for any . Note that , so, , and this quantity can be made sufficiently small. A similar argument applies when . And so is continous at any point.