Assume $f(x)\in C[0,+\infty)$，and for all $a\geqslant 0$, we have \begin{align*} \lim_{x\to \infty}(f(x+a)-f(x))=0 \tag{*}. \end{align*} Prove that there exists $g(x)\in C[0,+\infty)$ and $h(x)\in C^1[0,+\infty)$ such that $f(x)=g(x)+h(x)$, and such that they satisfy \begin{align*} \lim_{x\to \infty}g(x)=0,~~\lim_{x\to \infty}h'(x)=0. \end{align*}

My thought is let $h(x)=\frac1 a\int_x^{x+a}f(t)\,dt$, then it is easy to see $\lim_{x\to \infty}h'(x)=0$, but I can't explain that $\lim_{x\to \infty}g(x)=\lim_{x\to \infty}f(x)-h(x)=0$. It seems we should try proving $\lim_{x\to +\infty}f(x)$ exists by using the condition of (*), but I'm not sure whether it's true or false.