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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.

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