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.