# Profile decomposition of $f$ satisfying $\lim_{x\to\infty}[f(x+a)-f(x)]=0$, for any $a\geq 0$.

Let $$f\in C[0,\infty)$$, and $$\forall\ a\geq 0, \lim_{x\to\infty}[f(x+a)-f(x)]=0.$$ Show that there exists continuous $$g$$ and continously differentiable $$h$$, such that $$f=g+h$$,$$\lim_{x\to\infty}g(x)=0, \lim_{x\to\infty}h'(x)=0$$.

Let $$h(x)=\int_x^{x+1}f(t)dt$$, then $$\lim_{x\to\infty}h'(x)=0$$ simutanenously. But the property of $$g$$ maybe difficult to prove.