Assume $f$ and $g$ are continuous and related to each other as $ f(x) = \int _{0}^{x-1} \Big ( (x- y)^2 - 1\Big )^{3/2}g(y) \, dy, \qquad x>1. $
If we happen to know that $f$ is real analytic at some point $x_0$ can we deduce that there is a point $\phi (x_0)$ where $g$ is real analytic? (I suspect $\phi (x_0)$ should be one of the points $x_0 \pm 1$.)
I'm also very interested in incomplete answers such as possible ways to prove this.