Suppose $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is continuous. Also let $\mathbf{x}=(x_1,\ldots,x_d)\in\mathbb{R}^d$.
I'd like to prove the following:
If
$$\int_{\mathbb{R}^d}x_k\exp(-\mathbf{x}^T\mathbf{x})f(g(x))\,d\mathbf{x}=0$$
for all bounded continuous functions $f$, then $g(\mathbf{x})=g(-\mathbf{x})$ for all $\mathbf{x}\in\mathbb{R}^d$, that is $g$ is an even function.