Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ be continuous, compactly supported functions. Suppose that $f$ is supported on the interval $[0,1]$, and $g$ is constant on the interval $[0,2]$ (i.e., there is a real number $c$ such that $g(x) =c$ for all $x \in [0,2]$). Show that the convolution $f * g$ is constant on the interval $[1,2]$.
Let $f * g (x)= \int_{-\infty}^\infty f(y) g(x-y) dy$. $f(y) \not= 0$ for $y \in [0, 1]$, and $g(x-y) \not= 0$ for $y \in [x-2, x]$. Therefore, $f*g(x) \not= 0$ for $x \in [1, 2]$. But, I am not sure why the convolution should be constant. For example if $x=2$, $f * g (2)= \int_{0}^1 f(y) c dy$, and we don't know whether it is constant or not.