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It is known (see e.g. here [pdf]) that any function $f: \mathbb{R} -> \mathbb{R}$ with locally bounded left- and right derivatives can be represented as a difference of convex functions. Consequently, there must exist convex functions $f$ and $g$, such that the difference $h = f - g$ is (perhaps counter-intuitively) a highly oscillating function with arbitrary many local maxima and minima. My question is: Are there any nice explicit examples of functions $f, g$ and $h = f-g$ with these properties?

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  • $\begingroup$ Something like $\sin(nx)=(\sin(nx)+n^2x^2)-n^2x^2$ for any $n$? $\endgroup$
    – Gerd
    Commented Sep 13, 2023 at 8:04
  • $\begingroup$ Simple and does the job, thanks! You could post it as an answer. $\endgroup$
    – MKR
    Commented Sep 13, 2023 at 9:37
  • $\begingroup$ OK, I will post ist. $\endgroup$
    – Gerd
    Commented Sep 13, 2023 at 9:43

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For any $n \in \mathbb{N}$ the fucnctions $f(x)=\sin(nx)+n^2x^2$ and $g(x)=n^2x^2$ are convex (as the second derivative is $> 0$) and $\sin(nx)= f(x)-g(x)$ has arbitrarily many local maxima and minima.

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