I have some questions about the solution below, which appear after the solution.
Why can we assume that $\Delta f(a)$ and $\Delta f(b)$ are positive? If I directly plug in $g(x) = f(x) + c$ into the equation instead of f, I get $f(f(x)+c) + c)- f(x)-c = f(x)+c - x$, which may or may not hold because of the $f(f(x)+c)$ term.
Why is $\Delta f(f^{(m-1)}(b)) > \Delta f(a)$? I tried using a similar argument to what they discussed in the solutions but I still couldn't figure it out.