Consider a continuous but not differentiable function $f: \mathbb R \to \mathbb R$ and $Z$ a standard normal random variable. Define $g: \mathbb R \times \mathbb R_{>0} \to \mathbb R$ as the Gaussian kernel smoother $g(x,\epsilon) \equiv E[f(x + \epsilon \cdot Z)]$.
Are there known conditions on $f$ under which
- $g(x,\epsilon)$ is well-defined?
- $g(x,\epsilon)$ is differentiable in $x$?
- $\lim_{\epsilon \to 0} \frac{d}{dx} g(x,\epsilon)$ exists?
I think (1) just requires that $f$'s tails do not grow too quickly. (2) and (3) are intuitive to me, but I'm having trouble formalizing. Hoping to apply this in an economic model, using noise in a state variable's evolution to smooth out value functions of that state variable.