Answer to Question #164576 in Functional Analysis for Ebenezer

Question

Define f(x) = sinx on [0, 2pi]. Find two increasing functions h and g for which f = h — g on 

[0, 2pi]. 

Answer 1

We will search the functions $h,g:[0;2\pi]\to\mathbb{R}$ which are $\mathcal{C}^1$. We have $f'=(h-g)'=h'-g'$. The functions $h,g$ are strictly increasing if $h', g'>0$ on $[0;2\pi]$. Thus we need to find two positive continuous functions $h',g'$ such that $h'-g'=\cos x$. For $h'$ we need to take something that is always strictly bigger than $\cos x$, for example, $h'=2=const$. In this case $g'=2-\cos x$. Both are always positive. Now by finding their primitives $h=2x+C, g=2x-\sin(x)+C$, we see that the difference of such two functions is clearly $\sin(x)$ and they both are strictly increasing, as their derivatives are positive.