# fa.functional analysis – Finding the “root” of a monotone function (in the sense of composition)

Let $$f:(0,infty)rightarrow (0,infty)$$ be a smooth and monotone function s.t $$f(0)=0$$. Let $$Ninmathbb{N}$$. Can we find a function $$g: (0,infty) rightarrow (0,infty)$$ s.t $$gcirccdotscirc g$$ ($$g$$ composed with itself $$N$$ times) equals $$f$$?

Can we say something about $$g$$‘s monotonicity? Its smoothness? I cannot come up with any basic answers. Thanks in advance to the helpers.