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)=x
2, 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.
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
Part 1
