I want to show that the set of all the solutions to this functional equation $f(x-t) = f(x) -h$ where $f$ is strictly monotone continuous function is the set of affine functions $ax +b$. Update: The constants $t$ can be arbitrarily chosen, and $h$ is only dependent on $t$ and $f$.
Is my statement correct? And what would be a good approach to this problem? I am not sure how to tackle this problem without adding other assumptions such as $f$ is differentiable everywhere.