How about the function for , ? Clearly the function is differentiable at all non-zero , whereas1. Find a differentiable function f: R->R such that f'(0)=1 but f is not monotonically increasing on any interval (0,d).
for , making .
Furthermore if then .
Thus for all positive integers , showing that the function is not monotonic on any interval .
Of course, the function cannot be continuous at if it is to meet the conditions of the problem, wouldn't you say?