A problem in mathematical analysis that I have problems getting to grips with (need it to characterize the direction field of a differential equation) Suppose f(x) is a continuously differentiable function on ℝ with f(0)=f'(0)=0. Suppose that for any ε>0 there is some x in (0,ε) such that f(x)≥0. Does this mean that there is some δ>0 such that f(x)≥0 for all x in (0,δ)? If true, how to prove it? If it is not true, what would be a counterexample? What additional assumptions on f would make it true? I have been trying to prove this, but can't seem to discard the possibility of an oscillating f, with the oscillations taking place on smaller and smaller intervals and with lesser and lesser height as x=0 is approached. Still, intuition says that when x=0 is left, for example towards the right, f(x) must go somewhere.