How to proof the following function is always constant which satisfies $f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} $? Suppose that $f(x)$ is a bounded continuous function on $\mathbb{R}$,and that there exists a positive number $a$ such that
$$f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} $$
is constant. Can anybody show that  $f$ is necessarily constant ?

Write an equation that shows that the derivative of this expression is zero. This way you get an easy differential equation. Show that any solution gives a constant function.
