Suppose the following function :
$$f(x)=\frac{\theta}{2}g(-1)+(1-\theta)g(0)+\frac{\theta}{2}g(1),$$
where $g$ is some function.
Now it is written that if $g(-1)=g(1)$ and $g(0)=0$, then $f(x)=0$ for all $\theta$.
But it seems to me if $g(-1)=g(1)$ and $g(0)=0$, then $f(x)\ne 0$. Because if I specified some value, say, $g(-1)=g(1)=1$ and $g(0)=0$, then $f(x)=\theta$.
Where am I doing mistake?