Example of an infinitely differentiable function f : R → R with f(x) = 0 iff x = 0 and f intersects origin with infinite multiplicity

Question

Is there an infinitely differentiable function f : R → R with f(x) = 0 iff x = 0 for which it is reasonable to say the graph of f intersects the x-axis at the origin with infinite multiplicity. So y=x, y=x^2 does not count.

I think that considering the Taylor series of such a function would show that the desired function has to be identically zero, contrary to your hopes. — user795305, Oct 10 '14 at 01:57

score 3 · Accepted Answer · answered Oct 10 '14 at 02:02

3

$f(x) = \begin{cases} e^{-1/x^2}& x \neq 0 \cr 0 & x = 0\end{cases}$

looks like it does the trick. It's a standard proof by induction that it's smooth and that its Maclaurin series is identically zero.

answered Oct 10 '14 at 02:02

kahen

15,760

Example of an infinitely differentiable function f : R → R with f(x) = 0 iff x = 0 and f intersects origin with infinite multiplicity

1 Answers1