Let $X$ be a continuous real-valued random variable (for simplicity, we can assume $X\sim N(0,1)$) so that $P(X=a) = 0$ for some $a \in \mathbb{R}$. Let $f(x)$ be a nonconstant continuous (analytic or Lipschitz if needed) function such that $\mu = \mathbb{E}[f(X)] > 0$ and $0 < \text{Var}[f(X)] < \infty$.
I would like to know the conditions under which $P(f(X) = 0) = 0$. Or at least, would like to estimate how small it is, e.g. $P(f(X) \ne 0) > 1-\gamma$.
Obviously, if we already know $A=\{x | f(x) = 0\}$, $P(A)$ is the quantity of our interest. My attempt is as follows: Since $\mu > 0$, it follows from one of the concentration inequalities that $$ P(f(X) > 0) \ge 1 - \frac{\text{Var}[f(X)]}{\text{Var}[f(X)] + \mu^2} = \frac{(\mathbb{E}[f(X)])^2}{\mathbb{E}[f(X)^2]}, $$ which does not tell much...
I thought this would be easy to answer but it is not straightforward at all.
Any suggestions/comments/answers will be very appreciated. Thanks!
Denote by $g$ the density of $X$ and by $Z_f = \lbrace x\colon \, f(x)=0\rbrace$ the zero set of $f$. We have $$P(f(X)=0) = P(X \in Z_f) = \int_{Z_f} g(x)\, \mathrm{d} x.$$ Therefore, $P(f(X) =0) = 0$ is equivalent to: $$\text{for a.e. } x \in Z_f, \ g(x) =0.$$