continuous function

sam-n

$f$ is a continuous function on the non-negative reals. $f$ has the property that $f(x + a) - f(x) \to 0$ as $x \to \infty$ for each $a$ . Prove that we can find functions $g$ and $h$ such that: (1) $f(x) = g(x) + h(x)$ for all $x$ ; (2) $g(x) \to 0$ as $x \to \infty$ ; (3) $h$ is differentiable; and (4) $h'(x)\to 0$ as $x \to\infty$ .

Moderator edit: edited for readability

**Posted:** Tue Dec 28, 2004 11:09 am

Kent Merryfield

sam-n's post is showing up with a fair amount of garbage on my display, so I'll re-type what I think he said:

$f$ is a continuous function on the non-negative reals. $f$ has the property that $f(x + a) - f(x)\to 0$ as $x\to\infty$ for each $a.$ Prove that we can find functions $g$ and $h$ such that:
(1) $f(x) = g(x) + h(x)$ for all $x$ ;
(2) $g(x) \to0$ as $x\to\infty$ ;
(3) $h$ is differentiable; and
(4) $h'(x)\to0$ as $x\to\infty.$

**Posted:** Tue Dec 28, 2004 3:47 pm

harazi

Nice problem. I will take the function $h(x)=\frac{1}{b-a}\cdot\int_{a+x}^{b+x} {f(t)dt}$ for a certain a,b, good enough. Obviously, $h$ is derivable and its derivative tends to 0 as $x$ goes to infinity. Moreover, we have $h(x)-f(x)=\frac{1}{b-a}\cdot\int_{a}^{b} {f(t+x)-f(x) dt}$ . If we manage to find $a,b$ such that $f(t+x)-f(x)$ is bounded by an absolute constant for large $x$ and $t$ between $a$ and $b,$ we are done. But finding $a,b$ follows from Baire�s theorem, applied for the sets $A_n=\{a: |f(x+a)-f(x)|\leq \epsilon$ for all $x\geq n\}$ , which are closed and have union the whole set of real numbers. Thus we can find $n$ and $a,b$ such that $A_n$ contains $[a,b].$

**Posted:** Wed Dec 29, 2004 12:45 am