I have posted a similar question in the past but let me make a final try in a simpler framework.

Let $g \in C_0 ^\infty (\mathbb{R})$ be smooth and compactly supported. Define $$ f(x) = \int \big ((x - y)^2 - 1 \big )^{1/2}(x-y) g (y) \,dy $$ where integration is performed over the set where $|y - x|>1$ and $y\in \operatorname {supp}g$.

If $g$ fails to be real analytic at some point $x_0$ can we deduce that also $f$ fails to be real analytic at some point depending on $x_0$, like perhaps $x_0 \pm 1$?