# Questions about total bounded functions

Let $f(x)=\sin(x)$ on $x \in[0,2\pi]$. Find two increasing functions $h$ and $g$ such that $f=g-h$ on $x \in [0, 2\pi]$.

Finding the explicit example is where I'm stuck. Since this is a bounded function of finite tototal variation I know an explicit $h$ and $g$ exists. I just don't know what it is.

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## 1 Answer

Hint: $g$ can be any function for which $g'$ is large enough.

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I don't think I follow. – emka Dec 10 '12 at 9:10
@emka: If $f=g-h$, then $g'=f\,'+h'$; $|f\,'|$ is bounded, so if you take $h'$ big enough, you can ensure that both $g'$ and $h'$ are positive. What do you know about a function with a positive derivative? – Brian M. Scott Dec 10 '12 at 9:46
A function of positive derivative is increasing.  Could I take something simple and friendly like f(x)=x? – emka Dec 10 '12 at 13:40
You mean $g(x) = x$.  Yes you can (though you might prefer $2x$). – Robert Israel Dec 10 '12 at 20:25
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