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Suppose we have a strictly increasing, continuous scalar variable function $f(x)$ such that $f'(x) = 0 \text{ a.e.}$ The problem is $$\underset{x \in [a, b]}{\max}\ f(x) - cx$$ for some positive $c$. Since we have a sum of two continuous functions, it follows from the Weirstrass theorem that there exists a solution. If solution is interior, we cannot find it from the first order condition since $f'(x) = 0 \text{ a.e}$.

My question is: are there ways to characterize the interior solution analytically in general (without specifying $f(x)$)? I am also interested in a more general setting, where instead of $-cx$ there could be any continuous function.

If you have references to papers which study this, would be very grateful if you share.

D F
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    Interesting question. Obviously, you need to know something more about $f$ to have any chance of solving it. For a given $a, b, c, f(a), f(b)$, you can choose $\epsilon > 0$ and either $f$ constant on $[a+\epsilon, b]$ and to raise Cantor-style on $[a, a+\epsilon]$, or else choose $f$ constant on $[a, b-\epsilon]$ and raising Cantor-style on $[b-\epsilon, b]$ to have any maximum value in $(f(a) - cb, f(b) - ca)$. – Paul Sinclair Jan 19 '21 at 18:31
  • @PaulSinclair I mentioned strictly increasing function, so it will not be a constant on any interval. I understand that we can always find such functions where any maximum value is possible; I am interested in the ways of finding and characterizing this maximum value. For example, if f(x) is absolutely continuous it is also true that we can find a function for any possible maximum value, but for any such a function we have a way of characterizing a solution (FOC, SOC). I am wondering if there are some methods for the case I've described. – D F Jan 20 '21 at 04:44
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    "(FOC, SOC)" means? – Paul Sinclair Jan 20 '21 at 04:47
  • @PaulSinclair First/Second order conditions, sorry for that, probably I should have not used the abbreviations. – D F Jan 20 '21 at 05:32
  • Still not familar with what you are referring to. But that is an aside. Do you have a proof of existence for strictly increasing functions with $f' = 0$ a.e.? The Cantor function is mostly constant, and when I consider methods to address that, they do not converge to something with $f' = 0$ a.e. – Paul Sinclair Jan 20 '21 at 13:16
  • Never mind. I found one here. – Paul Sinclair Jan 20 '21 at 13:25

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