Let $f$ is even continuous function with compact support such that $\int f(t)dt=1 $, $g$ is bounded continuous function and convolution (f*g)(x)=g(x). How to prove that if $g$ has global minimum then $g$ is constant. I thought about Convolution Theorem but it seems it doesn't work in this case. May be without restriction that $g$ has global minimum there is a way to prove that $g$ is linear.