Smoothness of solutions to Fredholm integral equation

Question

Let $K(x,y)=k(|x-y|)$ where $k$ is continuous on $(0,1]$, and assume function $f\in L^2[0,1]$ satisfies $f(x)=\int_0^1 f(y)K(x,y)dy$. Is $f$ necessarily $C^\infty $ ? under what condition on kernel of integral equation solutions become smooth?What is a good reference to study more about Fredholm equations with singular kernel?