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.