Let $g,h$ be independent standard normal variables ($\cal{N}(0,1)$). Fix $\sigma>0$ and pick $f:\mathbb{R}\rightarrow \mathbb{R}$. Under what conditions on $f$, we have that
$$ \mathbb{E}[f(g+\sigma h)h]=\sigma \mathbb{E}[f(g+\sigma h)g] $$
I believe multivariate Stein's Lemma (https://en.wikipedia.org/wiki/Stein%27s_lemma) implies this property when $f$ is differentiable however I am interested in non-differentiable $f$ (specifically a step function).
