Counterexample to: If odd Taylor coefficients for g in a=0 equals 0, then g is an even function.

Question

Let g be an infinitely differentiable function.

Find counterexample to: If the odd Taylor coefficients for g in a=0 equals 0, then g is an even function.

Answer 1

There are infinitely differentiable functions which are not equal to their Taylor expansions (except at $x=0$). For instance:

$$ f(x) = \begin{cases} e^{-1/x^2} & \text{if $x>0$} \\ 0 & \text{if $x≤0$} \end{cases}$$

This has a Taylor expansion which is identically $0$ but the function is not even.