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Monotonic Function Decomposition (Posted on 2020-04-08) Difficulty: 3 of 5
Many real-valued functions f:R->R have a decomposition f=up+down into two strictly monotonic components, one increasing, one decreasing. For example, if f(x)=x2, one can choose up(x)=|x|max(x,0)+x and down(x)=|x|max(-x,0)-x.

1. Find a monotonic decomposition of the cosine.

2. Find a differentiable function f:R->R without such decomposition.

3. Which slightly stronger condition than differentiability ensures that a monotonic decomposition exists? Provide a formula for suitable up's and down's.

No Solution Yet Submitted by JLo    
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Some Thoughts Part 1 Comment 1 of 1
I think this ought to work:

   up(x) = max(cos(x),0) + x
   down(x) = min(cos(x),0) - x

Or even simpler:

   up(x) = cos(x) + x
   down(x) = -x

Note that for the second decomposition, the derivative of up(x) is occasionally zero, but it is still the case that if a > b then up(a) > up(b).

If the occasionally zero derivative bothers you, 
then you could just use
 
   up(x) = cos(x) + 2x
   down(x) = -2x


  Posted by Steve Herman on 2020-04-09 20:01:10
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