Find all real-valued continuous function f on the reals such that f(x)=f(cos x) for every real number x.

Find all real numbers \(\lambda\) and the corresponding functions \(f\) such that the equation

\(\displaystyle \int_0^1 \min(x,y) f(y) \,dy=\lambda f(x)\) has a non-zero solution \(f\) that is continuous on the interval [0,1].